![University Printing House, Cambridge CB2 8BS, United Kingdom
Cambridge University Press is part of the University of Cambridge.
It furthers the University’s mission by disseminating knowledge in the pursuit of
education, learning, and research at the highest international levels of excellence.
www.cambridge.org
Information on this title: www.cambridge.org/9781107552371
c
Cambridge University Press 2016
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.
First published 2016
Printed in the United Kingdom by Clays, St Ives plc
A catalogue record for this publication is available from the British Library.
Library of Congress Cataloging-in-Publication Data
Names: Durham Easter School (2014 : University of Durham) | Badziahin,
Dmitry (Dmitry A.), editor. | Gorodnik, Alexander,
1975– editor. | Peyerimhoff, Norbert, 1964– editor.
Title: Dynamics and analytic number theory : proceedings of the Durham Easter
School 2014 / edited by Dzmitry Badziahin, University of Durham, Alexander
Gorodnik, University of Bristol, Norbert Peyerimhoff, University of Durham.
Description: Cambridge : Cambridge University Press, [2016] |
Series: London Mathematical Society lecture note series ; 437 |
Includes bibliographical references and index.
Identifiers: LCCN 2016044609 | ISBN 9781107552371 (alk. paper)
Subjects: LCSH: Number theory – Congresses. | Dynamics – Congresses.
Classification: LCC QA241 .D87 2014 | DDC 512.7/3–dc23
LC record available at https://lccn.loc.gov/2016044609
ISBN 978-1-107-55237-1 Paperback
Cambridge University Press has no responsibility for the persistence or accuracy of
URLs for external or third-party Internet Web sites referred to in this publication
and does not guarantee that any content on such Web sites is, or will remain,
accurate or appropriate.](https://image.slidesharecdn.com/90206-250428142636-a7c29077/85/Dynamics-and-Analytic-Number-Theory-1st-Edition-Dzmitry-Badziahin-11-320.jpg)
![xii Preface
Chapter 3 is devoted to the study of exponential sums. Given a real
polynomial P(x) = ak xk + · · · + a1x + a0, the Weyl sums are defined as
WN :=
N−1
n=0
e2πi P(n)
.
The study of Weyl sums has a long history that goes back to foundational
works of Hardy, Littlewood, and Weyl. When the coefficients of the polyno-
mial P(X) satisfy a suitable irrationality condition, then it is known that for
some w ∈ (0, 1),
WN = O(N1−w
) as N → ∞,
and improving the value of the exponent in this estimate is a topic of current
research. This problem has been approached recently by several very different
methods. The method of Wooley is based on refinements of the Vinogradov
mean value theorem and a new idea of efficient congruencing, and the method
of Flaminio and Forni involves the investigation of asymptotic properties of
flows on nilmanifolds using renormalisation techniques. It is quite remarkable
that the exponents w obtained by the Flaminio–Forni approach, which is deter-
mined by optimal scaling of invariant distributions, essentially coincide with
the exponents derived by Wooley using his method of efficient congruencing.
As discussed in Chapter 3, flows on nilmanifolds provide a very convenient
tool for investigating the distribution of polynomial sequences modulo one and
modelling Weyl sums. We illustrate this by a simple example. Let
N :=
⎧
⎨
⎩
[p, q, r] :=
⎛
⎝
1 p r
0 1 q
0 0 1
⎞
⎠ : p, q,r ∈ R
⎫
⎬
⎭
denote the three-dimensional Heisenberg group, and be the subgroup con-
sisting of matrices with integral entries. Then the factor space M := N
provides the simplest example of a nilmanifold. Given an upper triangular
nilpotent matrix X = (xi j ), the flow generated by X is defined by
φX
t (m) = m exp(t X) with m ∈ M.
More explicitly, exp(t X) = [x12t, x23t, x13t + x12x23t2/2]. The space M con-
tains a two-dimensional subtorus T defined by the condition q = 0. If we
take x23 = 1, then the intersection of the orbit φX
t (e) with this torus gives
the sequence of points [x12n, 0, x13n + x12n2/2] with n ∈ N. Hence, choos-
ing suitable matrices X, the flows φX
t can be used to model values of general
quadratic polynomials P modulo one. Moreover, this relation can be made](https://image.slidesharecdn.com/90206-250428142636-a7c29077/85/Dynamics-and-Analytic-Number-Theory-1st-Edition-Dzmitry-Badziahin-19-320.jpg)
![Preface xiii
much more precise. In particular, with a suitable choice of a test function F on
M and m ∈ M,
N−1
n=0
e2πi P(n)
=
N
0
F(φX
t (m)) dt + O(1).
This demonstrates that quadratic Weyl sums are intimately related to averages
of one-parameter flows on the Heisenberg manifold. A more elaborate con-
struction discussed in detail in Chapter 3 shows that general Weyl sums can
be approximated by integrals along orbits on higher-dimensional nilmanifolds.
Chapter 3 discusses asymptotic behaviour of orbits averages on nilmanifolds
and related estimates for Weyl sums.
Dynamical systems techniques also provide powerful tools to analyse com-
binatorial structures of large subsets of integers and of more general groups.
This active research field fusing ideas from Ramsey Theory, Additive Combi-
natorics, and Ergodic Theory is surveyed in Chapter 4. We say that a subset
E ⊂ Z has positive upper density if
d̄(E) := lim sup
N−M→∞
|E ∩ [M, N]|
N − M
0.
Surprisingly, this soft analytic condition on the set E has profound combina-
torial consequences, one of the most remarkable of which is the Szemerédi
theorem. It states that every subset of positive density contains arbitrarily long
arithmetic progressions: namely, configurations of the form a, a + n, . . . , a +
(k − 1)n with arbitrary large k. It should be noted that the existence of three-
term arithmetic progressions had previously been established by Roth using
a variant of the circle method, but the case of general progressions required
substantial new ideas. Shortly after Szemerédi’s work appeared, Furstenberg
discovered a very different ingenious approach to this problem that used
ergodic-theoretic techniques. He realised that the Szemerédi theorem is equiv-
alent to a new ergodic-theoretic phenomenon called multiple recurrence. This
unexpected connection is summarised by the Furstenberg correspondence prin-
ciple which shows that, given a subset E ⊂ Z, one can construct a probability
space (X, μ), a measure-preserving transformation T : X → X, and a
measurable subset A ⊂ X such that μ(A) = d̄(E) and
d̄(E ∩(E −n)∩· · ·∩(E −(k −1)n)) ≥ μ(A ∩ T −n
(A)∩· · ·∩ T−(k−1)n
(A)).
This allows the proof of Szemerédi’s theorem to be reduced to establishing the
multiple recurrence property, which shows that if μ(A) 0 and k ≥ 1, then
there exists n ≥ 1 such that
μ(A ∩ T −n
A ∩ · · · ∩ T −(k−1)n
A) 0.](https://image.slidesharecdn.com/90206-250428142636-a7c29077/85/Dynamics-and-Analytic-Number-Theory-1st-Edition-Dzmitry-Badziahin-20-320.jpg)
Dynamics and Analytic Number Theory 1st Edition Dzmitry Badziahin
- 1. Dynamics and Analytic Number Theory 1st Edition Dzmitry Badziahin download https://ebookultra.com/download/dynamics-and-analytic-number- theory-1st-edition-dzmitry-badziahin/ Explore and download more ebooks or textbooks at ebookultra.com
- 2. We have selected some products that you may be interested in Click the link to download now or visit ebookultra.com for more options!. Duality in Analytic Number Theory 1st Edition Peter D. T. A. Elliott https://ebookultra.com/download/duality-in-analytic-number-theory-1st- edition-peter-d-t-a-elliott/ Algorithmic number theory lattices number fields curves and cryptography 1st Edition J.P. Buhler https://ebookultra.com/download/algorithmic-number-theory-lattices- number-fields-curves-and-cryptography-1st-edition-j-p-buhler/ Fluid Transport Theory Dynamics and Applications Theory Dynamics and Applications 1st Edition Emma T. Berg https://ebookultra.com/download/fluid-transport-theory-dynamics-and- applications-theory-dynamics-and-applications-1st-edition-emma-t-berg/ Recurrence in Ergodic Theory and Combinatorial Number Theory Harry Furstenberg https://ebookultra.com/download/recurrence-in-ergodic-theory-and- combinatorial-number-theory-harry-furstenberg/
- 3. Elliptic Tales Curves Counting and Number Theory Avner Ash https://ebookultra.com/download/elliptic-tales-curves-counting-and- number-theory-avner-ash/ Number Theory Fourier Analysis and Geometric Discrepancy London Mathematical Society Student Texts Series Number 81 1st Edition Giancarlo Travaglini https://ebookultra.com/download/number-theory-fourier-analysis-and- geometric-discrepancy-london-mathematical-society-student-texts- series-number-81-1st-edition-giancarlo-travaglini/ Elementary Number Theory with Applications 2nd Edition Koshy https://ebookultra.com/download/elementary-number-theory-with- applications-2nd-edition-koshy/ Combinatorics and Number Theory of Counting Sequences 1st Edition Istvan Mezo (Author) https://ebookultra.com/download/combinatorics-and-number-theory-of- counting-sequences-1st-edition-istvan-mezo-author/ Current Trends in Number Theory 1st Edition Sukumar Das Adhikari https://ebookultra.com/download/current-trends-in-number-theory-1st- edition-sukumar-das-adhikari/
- 5. Dynamics and Analytic Number Theory 1st Edition Dzmitry Badziahin Digital Instant Download Author(s): Dzmitry Badziahin, Alexander Gorodnik, Norbert Peyerimhoff (eds.) ISBN(s): 9781107552371, 1107552370 Edition: 1 File Details: PDF, 1.88 MB Year: 2017 Language: english
- 8. LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor: Professor M. Reid, Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom The titles below are available from booksellers, or from Cambridge University Press at http://www.cambridge.org/mathematics 314 Spectral generalizations of line graphs, D. CVETKOVIĆ, P. ROWLINSON & S. SIMIĆ 315 Structured ring spectra, A. BAKER & B. RICHTER (eds) 316 Linear logic in computer science, T. EHRHARD, P. RUET, J.-Y. GIRARD & P. SCOTT (eds) 317 Advances in elliptic curve cryptography, I.F. BLAKE, G. SEROUSSI & N.P. SMART (eds) 318 Perturbation of the boundary in boundary-value problems of partial differential equations, D. HENRY 319 Double affine Hecke algebras, I. CHEREDNIK 320 L-functions and Galois representations, D. BURNS, K. BUZZARD & J. NEKOVÁŘ (eds) 321 Surveys in modern mathematics, V. PRASOLOV & Y. ILYASHENKO (eds) 322 Recent perspectives in random matrix theory and number theory, F. MEZZADRI & N.C. SNAITH (eds) 323 Poisson geometry, deformation quantisation and group representations, S. GUTT et al (eds) 324 Singularities and computer algebra, C. LOSSEN & G. PFISTER (eds) 325 Lectures on the Ricci flow, P. TOPPING 326 Modular representations of finite groups of Lie type, J.E. HUMPHREYS 327 Surveys in combinatorics 2005, B.S. WEBB (ed) 328 Fundamentals of hyperbolic manifolds, R. CANARY, D. EPSTEIN & A. MARDEN (eds) 329 Spaces of Kleinian groups, Y. MINSKY, M. SAKUMA & C. SERIES (eds) 330 Noncommutative localization in algebra and topology, A. RANICKI (ed) 331 Foundations of computational mathematics, Santander 2005, L.M PARDO, A. PINKUS, E. SÜLI & M.J. TODD (eds) 332 Handbook of tilting theory, L. ANGELERI HÜGEL, D. HAPPEL & H. KRAUSE (eds) 333 Synthetic differential geometry (2nd Edition), A. KOCK 334 The Navier–Stokes equations, N. RILEY & P. DRAZIN 335 Lectures on the combinatorics of free probability, A. NICA & R. SPEICHER 336 Integral closure of ideals, rings, and modules, I. SWANSON & C. HUNEKE 337 Methods in Banach space theory, J.M.F. CASTILLO & W.B. JOHNSON (eds) 338 Surveys in geometry and number theory, N. YOUNG (ed) 339 Groups St Andrews 2005 I, C.M. CAMPBELL, M.R. QUICK, E.F. ROBERTSON & G.C. SMITH (eds) 340 Groups St Andrews 2005 II, C.M. CAMPBELL, M.R. QUICK, E.F. ROBERTSON & G.C. SMITH (eds) 341 Ranks of elliptic curves and random matrix theory, J.B. CONREY, D.W. FARMER, F. MEZZADRI & N.C. SNAITH (eds) 342 Elliptic cohomology, H.R. MILLER & D.C. RAVENEL (eds) 343 Algebraic cycles and motives I, J. NAGEL & C. PETERS (eds) 344 Algebraic cycles and motives II, J. NAGEL & C. PETERS (eds) 345 Algebraic and analytic geometry, A. NEEMAN 346 Surveys in combinatorics 2007, A. HILTON & J. TALBOT (eds) 347 Surveys in contemporary mathematics, N. YOUNG & Y. CHOI (eds) 348 Transcendental dynamics and complex analysis, P.J. RIPPON & G.M. STALLARD (eds) 349 Model theory with applications to algebra and analysis I, Z. CHATZIDAKIS, D. MACPHERSON, A. PILLAY & A. WILKIE (eds) 350 Model theory with applications to algebra and analysis II, Z. CHATZIDAKIS, D. MACPHERSON, A. PILLAY & A. WILKIE (eds) 351 Finite von Neumann algebras and masas, A.M. SINCLAIR & R.R. SMITH 352 Number theory and polynomials, J. MCKEE & C. SMYTH (eds) 353 Trends in stochastic analysis, J. BLATH, P. MÖRTERS & M. SCHEUTZOW (eds) 354 Groups and analysis, K. TENT (ed) 355 Non-equilibrium statistical mechanics and turbulence, J. CARDY, G. FALKOVICH & K. GAWEDZKI 356 Elliptic curves and big Galois representations, D. DELBOURGO 357 Algebraic theory of differential equations, M.A.H. MACCALLUM & A.V. MIKHAILOV (eds) 358 Geometric and cohomological methods in group theory, M.R. BRIDSON, P.H. KROPHOLLER & I.J. LEARY (eds) 359 Moduli spaces and vector bundles, L. BRAMBILA-PAZ, S.B. BRADLOW, O. GARCÍA-PRADA & S. RAMANAN (eds) 360 Zariski geometries, B. ZILBER 361 Words: Notes on verbal width in groups, D. SEGAL 362 Differential tensor algebras and their module categories, R. BAUTISTA, L. SALMERÓN & R. ZUAZUA 363 Foundations of computational mathematics, Hong Kong 2008, F. CUCKER, A. PINKUS & M.J. TODD (eds) 364 Partial differential equations and fluid mechanics, J.C. ROBINSON & J.L. RODRIGO (eds) 365 Surveys in combinatorics 2009, S. HUCZYNSKA, J.D. MITCHELL & C.M. RONEY-DOUGAL (eds) 366 Highly oscillatory problems, B. ENGQUIST, A. FOKAS, E. HAIRER & A. ISERLES (eds) 367 Random matrices: High dimensional phenomena, G. BLOWER 368 Geometry of Riemann surfaces, F.P. GARDINER, G. GONZÁLEZ-DIEZ & C. KOUROUNIOTIS (eds) 369 Epidemics and rumours in complex networks, M. DRAIEF & L. MASSOULIÉ 370 Theory of p-adic distributions, S. ALBEVERIO, A.YU. KHRENNIKOV & V.M. SHELKOVICH 371 Conformal fractals, F. PRZYTYCKI & M. URBAŃSKI 372 Moonshine: The first quarter century and beyond, J. LEPOWSKY, J. MCKAY & M.P. TUITE (eds) 373 Smoothness, regularity and complete intersection, J. MAJADAS & A. G. RODICIO 374 Geometric analysis of hyperbolic differential equations: An introduction, S. ALINHAC 375 Triangulated categories, T. HOLM, P. JØRGENSEN & R. ROUQUIER (eds)
- 9. 376 Permutation patterns, S. LINTON, N. RUŠKUC & V. VATTER (eds) 377 An introduction to Galois cohomology and its applications, G. BERHUY 378 Probability and mathematical genetics, N. H. BINGHAM & C. M. GOLDIE (eds) 379 Finite and algorithmic model theory, J. ESPARZA, C. MICHAUX & C. STEINHORN (eds) 380 Real and complex singularities, M. MANOEL, M.C. ROMERO FUSTER & C.T.C WALL (eds) 381 Symmetries and integrability of difference equations, D. LEVI, P. OLVER, Z. THOMOVA & P. WINTERNITZ (eds) 382 Forcing with random variables and proof complexity, J. KRAJÍČEK 383 Motivic integration and its interactions with model theory and non-Archimedean geometry I, R. CLUCKERS, J. NICAISE & J. SEBAG (eds) 384 Motivic integration and its interactions with model theory and non-Archimedean geometry II, R. CLUCKERS, J. NICAISE & J. SEBAG (eds) 385 Entropy of hidden Markov processes and connections to dynamical systems, B. MARCUS, K. PETERSEN & T. WEISSMAN (eds) 386 Independence-friendly logic, A.L. MANN, G. SANDU & M. SEVENSTER 387 Groups St Andrews 2009 in Bath I, C.M. CAMPBELL et al (eds) 388 Groups St Andrews 2009 in Bath II, C.M. CAMPBELL et al (eds) 389 Random fields on the sphere, D. MARINUCCI & G. PECCATI 390 Localization in periodic potentials, D.E. PELINOVSKY 391 Fusion systems in algebra and topology, M. ASCHBACHER, R. KESSAR & B. OLIVER 392 Surveys in combinatorics 2011, R. CHAPMAN (ed) 393 Non-abelian fundamental groups and Iwasawa theory, J. COATES et al (eds) 394 Variational problems in differential geometry, R. BIELAWSKI, K. HOUSTON & M. SPEIGHT (eds) 395 How groups grow, A. MANN 396 Arithmetic differential operators over the p-adic integers, C.C. RALPH & S.R. SIMANCA 397 Hyperbolic geometry and applications in quantum chaos and cosmology, J. BOLTE & F. STEINER (eds) 398 Mathematical models in contact mechanics, M. SOFONEA & A. MATEI 399 Circuit double cover of graphs, C.-Q. ZHANG 400 Dense sphere packings: a blueprint for formal proofs, T. HALES 401 A double Hall algebra approach to affine quantum Schur–Weyl theory, B. DENG, J. DU & Q. FU 402 Mathematical aspects of fluid mechanics, J.C. ROBINSON, J.L. RODRIGO & W. SADOWSKI (eds) 403 Foundations of computational mathematics, Budapest 2011, F. CUCKER, T. KRICK, A. PINKUS & A. SZANTO (eds) 404 Operator methods for boundary value problems, S. HASSI, H.S.V. DE SNOO & F.H. SZAFRANIEC (eds) 405 Torsors, étale homotopy and applications to rational points, A.N. SKOROBOGATOV (ed) 406 Appalachian set theory, J. CUMMINGS & E. SCHIMMERLING (eds) 407 The maximal subgroups of the low-dimensional finite classical groups, J.N. BRAY, D.F. HOLT & C.M. RONEY-DOUGAL 408 Complexity science: the Warwick master’s course, R. BALL, V. KOLOKOLTSOV & R.S. MACKAY (eds) 409 Surveys in combinatorics 2013, S.R. BLACKBURN, S. GERKE & M. WILDON (eds) 410 Representation theory and harmonic analysis of wreath products of finite groups, T. CECCHERINI-SILBERSTEIN, F. SCARABOTTI & F. TOLLI 411 Moduli spaces, L. BRAMBILA-PAZ, O. GARCÍA-PRADA, P. NEWSTEAD & R.P. THOMAS (eds) 412 Automorphisms and equivalence relations in topological dynamics, D.B. ELLIS & R. ELLIS 413 Optimal transportation, Y. OLLIVIER, H. PAJOT & C. VILLANI (eds) 414 Automorphic forms and Galois representations I, F. DIAMOND, P.L. KASSAEI & M. KIM (eds) 415 Automorphic forms and Galois representations II, F. DIAMOND, P.L. KASSAEI & M. KIM (eds) 416 Reversibility in dynamics and group theory, A.G. O’FARRELL & I. SHORT 417 Recent advances in algebraic geometry, C.D. HACON, M. MUSTAŢĂ & M. POPA (eds) 418 The Bloch–Kato conjecture for the Riemann zeta function, J. COATES, A. RAGHURAM, A. SAIKIA & R. SUJATHA (eds) 419 The Cauchy problem for non-Lipschitz semi-linear parabolic partial differential equations, J.C. MEYER & D.J. NEEDHAM 420 Arithmetic and geometry, L. DIEULEFAIT et al (eds) 421 O-minimality and Diophantine geometry, G.O. JONES & A.J. WILKIE (eds) 422 Groups St Andrews 2013, C.M. CAMPBELL et al (eds) 423 Inequalities for graph eigenvalues, Z. STANIĆ 424 Surveys in combinatorics 2015, A. CZUMAJ et al (eds) 425 Geometry, topology and dynamics in negative curvature, C.S. ARAVINDA, F.T. FARRELL & J.-F. LAFONT (eds) 426 Lectures on the theory of water waves, T. BRIDGES, M. GROVES & D. NICHOLLS (eds) 427 Recent advances in Hodge theory, M. KERR & G. PEARLSTEIN (eds) 428 Geometry in a Fréchet context, C. T. J. DODSON, G. GALANIS & E. VASSILIOU 429 Sheaves and functions modulo p, L. TAELMAN 430 Recent progress in the theory of the Euler and Navier-Stokes equations, J.C. ROBINSON, J.L. RODRIGO, W. SADOWSKI & A. VIDAL-LÓPEZ (eds) 431 Harmonic and subharmonic function theory on the real hyperbolic ball, M. STOLL 432 Topics in graph automorphisms and reconstruction (2nd Edition), J. LAURI & R. SCAPELLATO 433 Regular and irregular holonomic D-modules, M. KASHIWARA & P. SCHAPIRA 434 Analytic semigroups and semilinear initial boundary value problems (2nd Edition), K. TAIRA 435 Graded rings and graded Grothendieck groups, R. HAZRAT 436 Groups, graphs and random walks, T. CECCHERINI-SILBERSTEIN, M. SALVATORI & E. SAVA-HUSS (eds) 437 Dynamics and analytic number theory, D. BADZIAHIN, A. GORODNIK & N. PEYERIMHOFF (eds)
- 10. London Mathematical Society Lecture Note Series: 437 Dynamics and Analytic Number Theory Proceedings of the Durham Easter School 2014 Edited by DZMITRY BADZIAHIN University of Durham ALEXANDER GORODNIK University of Bristol NORBERT PEYERIMHOFF University of Durham
- 11. University Printing House, Cambridge CB2 8BS, United Kingdom Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107552371 c Cambridge University Press 2016 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2016 Printed in the United Kingdom by Clays, St Ives plc A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: Durham Easter School (2014 : University of Durham) | Badziahin, Dmitry (Dmitry A.), editor. | Gorodnik, Alexander, 1975– editor. | Peyerimhoff, Norbert, 1964– editor. Title: Dynamics and analytic number theory : proceedings of the Durham Easter School 2014 / edited by Dzmitry Badziahin, University of Durham, Alexander Gorodnik, University of Bristol, Norbert Peyerimhoff, University of Durham. Description: Cambridge : Cambridge University Press, [2016] | Series: London Mathematical Society lecture note series ; 437 | Includes bibliographical references and index. Identifiers: LCCN 2016044609 | ISBN 9781107552371 (alk. paper) Subjects: LCSH: Number theory – Congresses. | Dynamics – Congresses. Classification: LCC QA241 .D87 2014 | DDC 512.7/3–dc23 LC record available at https://lccn.loc.gov/2016044609 ISBN 978-1-107-55237-1 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate.
- 12. Contents List of contributors page vii Preface ix 1 Metric Diophantine Approximation: Aspects of Recent Work 1 Victor Beresnevich, Felipe Ramírez and Sanju Velani 1.1 Background: Dirichlet and Bad 1 1.2 Metric Diophantine Approximation: The Classical Lebesgue Theory 9 1.3 Metric Diophantine Approximation: The Classical Hausdorff Theory 18 1.4 The Higher-Dimensional Theory 27 1.5 Ubiquitous Systems of Points 46 1.6 Diophantine Approximation on Manifolds 53 1.7 The Badly Approximable Theory 74 2 Exponents of Diophantine Approximation 96 Yann Bugeaud 2.1 Introduction and Generalities 96 2.2 Further Definitions and First Results 99 2.3 Overview of Known Relations Between Exponents 106 2.4 Bounds for the Exponents of Approximation 109 2.5 Spectra 114 2.6 Intermediate Exponents 121 2.7 Parametric Geometry of Numbers 124 2.8 Real Numbers Which Are Badly Approximable by Algebraic Numbers 126 2.9 Open Problems 127 v
- 13. vi Contents 3 Effective Equidistribution of Nilflows and Bounds on Weyl Sums 136 Giovanni Forni 3.1 Introduction 137 3.2 Nilflows and Weyl Sums 142 3.3 The Cohomological Equation 152 3.4 The Heisenberg Case 159 3.5 Higher-Step Filiform Nilflows 174 4 Multiple Recurrence and Finding Patterns in Dense Sets 189 Tim Austin 4.1 Szemerédi’s Theorem and Its Relatives 189 4.2 Multiple Recurrence 192 4.3 Background from Ergodic Theory 197 4.4 Multiple Recurrence in Terms of Self-Joinings 212 4.5 Weak Mixing 222 4.6 Roth’s Theorem 230 4.7 Towards Convergence in General 238 4.8 Sated Systems and Pleasant Extensions 242 4.9 Further Reading 248 5 Diophantine Problems and Homogeneous Dynamics 258 Manfred Einsiedler and Tom Ward 5.1 Equidistribution and the Gauss Circle Problem 258 5.2 Counting Points in SL2(Z) · i ⊆ H 267 5.3 Dirichlet’s Theorem and Dani’s Correspondence 278 6 Applications of Thin Orbits 289 Alex Kontorovich 6.1 Lecture 1: Closed Geodesics, Binary Quadratic Forms, and Duke’s Theorem 289 6.2 Lecture 2: Three Problems in Continued Fractions: ELMV, McMullen, and Zaremba 306 6.3 Lecture 3: The Thin Orbits Perspective 310 Index 318
- 14. Contributors Tim Austin Courant Institute, NYU New York, NY 10012, USA Email: tim@cims.nyu.edu Victor Beresnevich Department of Mathematics, University of York Heslington, York, Y010 5DD, United Kingdom Email: victor.beresnevich@york.ac.uk Yann Bugeaud Département de mathématiques, Université de Strasbourg F-67084 Strasbourg, France Email: bugeaud@math.unistra.fr Manfred Einsiedler Departement Mathematik, ETH Zürich 8092 Zürich, Switzerland Email: manfred.einsiedler@math.ethz.ch Giovanni Forni Department of Mathematics, University of Maryland College Park, MD 20742-4015, USA Email: gforni@math.umd.edu Alex Kontorovich Department of Mathematics, Rutgers University Piscataway, NJ 08854, USA Email: alex.kontorovich@rutgers.edu vii
- 15. viii List of contributors Felipe Ramírez Department of Mathematics and Computer Science Wesleyan University, Middletown, CT 06459, USA Email: framirez@wesleyan.edu Sanju Velani Department of Mathematics, University of York Heslington, York, Y010 5DD, United Kingdom Email: slv3@york.ac.uk Tom Ward Executive Office, Palatine Centre, Durham University Durham DH1 3LE, United Kingdom Email: t.b.ward@durham.ac.uk
- 16. Preface This book is devoted to some of the interesting recently discovered interac- tions between Analytic Number Theory and the Theory of Dynamical Systems. Analytical Number Theory has a very long history. Many people associate its starting point with the work of Dirichet on L-functions in 1837, where he proved his famous result about infinitely many primes in arithmetic progres- sions. Since then, analytical methods have played a crucial role in proving many important results in Number Theory. For example, the study of the Riemann zeta function allowed to uncover deep information about the distri- bution of prime numbers. Hardy and Littlewood developed their circle method to establish first explicit general estimates for the Waring problem. Later, Vinogradov used the idea of the circle method to create his own method of exponential sums which allowed him to solve, unconditionally of the Rie- mann hypothesis, the ternary Goldbach conjecture for all but finitely many natural numbers. Roth also used exponential sums to prove the existence of three-term arithmetic progressions in subsets of positive density. One of the fundamental questions which arise in the investigation of exponential sums, as well as many other problems in Number Theory, is how rational numbers/vec- tors are distributed and how well real numbers/vectors can be approximated by rationals. Understanding various properties of sets of numbers/vectors that have prescribed approximational properties, such as their size, is the subject of the metric theory of Diophantine approximation, which involves an inter- esting interplay between Arithmetic and Measure Theory. While these topics are now considered as classical, the behaviour of exponential sums is still not well understood today, and there are still many challenging open problems in Diophantine approximation. On the other hand, in the last decades there have been several important breakthroughs in these areas of Number Theory where progress on long-standing open problems has been achieved by utilising techniques which originated from the Theory of Dynamical Systems. These ix
- 17. x Preface developments have uncovered many profound and very promising connections between number-theoretic and dynamical objects that are at the forefront of current research. For instance, it turned out that properties of exponential sums are intimately related to the behaviour of orbits of flows on nilmanifolds; the existence of given combinatorial configurations (e.g. arithmetic progressions) in subsets of integers can be established through the study of multiple recur- rence properties for dynamical systems; and Diophantine properties of vectors in the Euclidean spaces can be characterised in terms of excursions of orbits of suitable flows in the space of lattices. The material of this book is based on the Durham Easter School, ‘Dynam- ics and Analytic Number Theory’, that was held at the University of Durham in Spring 2014. The intention of this school was to communicate some of these remarkable developments at the interface between Number Theory and Dynamical Systems to young researchers. The Easter School consisted of a series of mini-courses (with two to three lectures each) given by Tim Austin, Manfred Einsiedler, Giovanni Forni, Alex Kontorovich, Sanju Velani and Trevor Wooley, and a talk by Yann Bugeaud presenting a collection of recent results and open problems in Diophantine approximation. The event was very well received by more than 60 participants, many of them PhD students from all around the world. Because of the great interest of young researchers in this topic, we decided to encourage the speakers to write contributions to this Proceedings volume. One of the typical examples where both classical and dynamical approaches are now actively developing and producing deep results is the theory of Dio- phantine approximation. One of the classical problems in this area asks how well a given n-dimensional vector x ∈ Rn can be approximated by vectors with rational coefficients. More specifically, one can ask: what is the supremum λ(x) of the values λ such that the inequality ||qx − p||∞ Q−λ (1) has infinitely many integer solutions Q ∈ N, q ∈ N, p ∈ Zn satisfying q ≤ Q? This type of problem is referred to as a simultaneous Diophantine approxima- tion. There is also a dual Diophantine approximation problem which asks for the supremum ω(x) of the values ω such that the inequality |(x, q) − p| Q−ω (2) has infinitely many solutions Q ∈ N, q ∈ Zn, p ∈ Z with q = 0 and ||q||∞ ≤ Q. It turns out that there are various relations between the exponents λ(x) and ω(x). Chapter 2 provides an overview of known relations between these and some other similar exponents. It mostly concentrates on the case where x lies
- 18. Preface xi on the so-called Veronese curve which is defined by x(t) := (t, t2, . . . , tn) with real t. This case is of particular importance for number theorists since it has implications for the question about the distribution of algebraic numbers of bounded degree. For example, condition (2) in this case transforms to |P(t)| Q−ω where P(t) is a polynomial with integer coefficients. For large Q this implies that x is very close to the root of P, which is an algebraic number. Metric theory of Diophantine approximation does not work with particular vectors x. Instead it deals with the sets of all vectors x satisfying inequali- ties like (1) or (2) for infinitely many Q ∈ N, q ∈ N, p ∈ Zn (respectively, Q ∈ N, q ∈ Zn, p ∈ Z, q = 0). The central problem is to estimate the mea- sure and the Hausdorff dimension of such sets. This area of Number Theory was founded at the beginning of the twentieth century with Khintchine’s work which was later generalised by Groshev. In the most general way they showed that, given a function ψ : R≥0 → R≥0, the set of m × n matrices A which satisfy the inequality ||Aq − p||∞ ψ(||q||∞) with p ∈ Zn and q ∈ Zm, has either zero or full Lebesgue measure. The matrices A satisfying this property are usually called ψ-well approximable. Furthermore, with some mild conditions on ψ, the Lebesgue measure of the set of ψ-well approximable matrices is determined by the convergence of a certain series which involves ψ. Later, many other results of this type were established, some of them with help of the classical methods and others by using the ideas from homogeneous dynamics. Chapter 1 describes several powerful ‘classical’ techniques used in metric theory of Diophantine approximation, such as the Mass Transference Prin- ciple, ubiquitous systems, Cantor sets constructions and winning sets. The Mass Transference Principle allows us to get results about the more sensi- tive Hausdorff measure and Hausdorff dimension of sets of well approximable matrices or similar objects as soon as results about their Lebesgue measure are known. Ubiquitous systems provide another powerful method originat- ing from works of A. Baker and W. Schmidt. It enables us to obtain the ‘full Lebesgue measure’-type results in various analogues of the Khintchine– Groshev theorem. Finally, Chapter 1 introduces the generalised Cantor set construction technique, which helps in investigating badly approximable num- bers or vectors. It also relates such sets with so-called winning sets developed by W. Schmidt. The winning sets have several surprising properties. For example, they have the maximal possible Hausdorff dimension and, even though such sets may be null in terms of Lebesgue measure, their countable intersection must also be winning.
- 19. xii Preface Chapter 3 is devoted to the study of exponential sums. Given a real polynomial P(x) = ak xk + · · · + a1x + a0, the Weyl sums are defined as WN := N−1 n=0 e2πi P(n) . The study of Weyl sums has a long history that goes back to foundational works of Hardy, Littlewood, and Weyl. When the coefficients of the polyno- mial P(X) satisfy a suitable irrationality condition, then it is known that for some w ∈ (0, 1), WN = O(N1−w ) as N → ∞, and improving the value of the exponent in this estimate is a topic of current research. This problem has been approached recently by several very different methods. The method of Wooley is based on refinements of the Vinogradov mean value theorem and a new idea of efficient congruencing, and the method of Flaminio and Forni involves the investigation of asymptotic properties of flows on nilmanifolds using renormalisation techniques. It is quite remarkable that the exponents w obtained by the Flaminio–Forni approach, which is deter- mined by optimal scaling of invariant distributions, essentially coincide with the exponents derived by Wooley using his method of efficient congruencing. As discussed in Chapter 3, flows on nilmanifolds provide a very convenient tool for investigating the distribution of polynomial sequences modulo one and modelling Weyl sums. We illustrate this by a simple example. Let N := ⎧ ⎨ ⎩ [p, q, r] := ⎛ ⎝ 1 p r 0 1 q 0 0 1 ⎞ ⎠ : p, q,r ∈ R ⎫ ⎬ ⎭ denote the three-dimensional Heisenberg group, and be the subgroup con- sisting of matrices with integral entries. Then the factor space M := N provides the simplest example of a nilmanifold. Given an upper triangular nilpotent matrix X = (xi j ), the flow generated by X is defined by φX t (m) = m exp(t X) with m ∈ M. More explicitly, exp(t X) = [x12t, x23t, x13t + x12x23t2/2]. The space M con- tains a two-dimensional subtorus T defined by the condition q = 0. If we take x23 = 1, then the intersection of the orbit φX t (e) with this torus gives the sequence of points [x12n, 0, x13n + x12n2/2] with n ∈ N. Hence, choos- ing suitable matrices X, the flows φX t can be used to model values of general quadratic polynomials P modulo one. Moreover, this relation can be made
- 20. Preface xiii much more precise. In particular, with a suitable choice of a test function F on M and m ∈ M, N−1 n=0 e2πi P(n) = N 0 F(φX t (m)) dt + O(1). This demonstrates that quadratic Weyl sums are intimately related to averages of one-parameter flows on the Heisenberg manifold. A more elaborate con- struction discussed in detail in Chapter 3 shows that general Weyl sums can be approximated by integrals along orbits on higher-dimensional nilmanifolds. Chapter 3 discusses asymptotic behaviour of orbits averages on nilmanifolds and related estimates for Weyl sums. Dynamical systems techniques also provide powerful tools to analyse com- binatorial structures of large subsets of integers and of more general groups. This active research field fusing ideas from Ramsey Theory, Additive Combi- natorics, and Ergodic Theory is surveyed in Chapter 4. We say that a subset E ⊂ Z has positive upper density if d̄(E) := lim sup N−M→∞ |E ∩ [M, N]| N − M 0. Surprisingly, this soft analytic condition on the set E has profound combina- torial consequences, one of the most remarkable of which is the Szemerédi theorem. It states that every subset of positive density contains arbitrarily long arithmetic progressions: namely, configurations of the form a, a + n, . . . , a + (k − 1)n with arbitrary large k. It should be noted that the existence of three- term arithmetic progressions had previously been established by Roth using a variant of the circle method, but the case of general progressions required substantial new ideas. Shortly after Szemerédi’s work appeared, Furstenberg discovered a very different ingenious approach to this problem that used ergodic-theoretic techniques. He realised that the Szemerédi theorem is equiv- alent to a new ergodic-theoretic phenomenon called multiple recurrence. This unexpected connection is summarised by the Furstenberg correspondence prin- ciple which shows that, given a subset E ⊂ Z, one can construct a probability space (X, μ), a measure-preserving transformation T : X → X, and a measurable subset A ⊂ X such that μ(A) = d̄(E) and d̄(E ∩(E −n)∩· · ·∩(E −(k −1)n)) ≥ μ(A ∩ T −n (A)∩· · ·∩ T−(k−1)n (A)). This allows the proof of Szemerédi’s theorem to be reduced to establishing the multiple recurrence property, which shows that if μ(A) 0 and k ≥ 1, then there exists n ≥ 1 such that μ(A ∩ T −n A ∩ · · · ∩ T −(k−1)n A) 0.
- 21. Random documents with unrelated content Scribd suggests to you:
- 22. Lieut. Brownsworth. Lieut. Edmund Brownsworth, Leicestershire Regiment, youngest son of the late Mr. David Brownsworth, artist, and Mrs. Brownsworth, of Skipton, died in the 10th Casualty Clearing Station at Abule, on May 29th, 1916. He received the first rudiments of education under Mr. Alfred Hartley at the Skipton National School and, later, after a period of training, was apprenticed to the Merchant Service. After many extended cruises, he left the sea on account of ill-health. Later, he joined the army and rose to the rank of Sergeant in the Leicestershire Regiment, and in that capacity went out to France in November, 1914. He so distinguished himself that the Brigadier- General recommended him for a commission. He was gazetted to his old regiment in April, 1915, as a 2nd Lieut., being advanced in November to a Temporary Lieutenancy.
- 23. Lieut. Slingsby. Lieut. Stephen Slingsby, H.M.S. “Defence,” fourth son of Mr. J. A. Slingsby and Mrs. Slingsby, Carla Beck, Carleton, was killed in action in the North Sea in June, 1916. The deceased officer was born on the 20th June, 1892, and was educated at St. Edmund’s School, Hindhead, and underwent training for his career in the Navy at the Royal Naval College at Osborne and Dartmouth, and on H.M.S. “Cornwall.” His ships included the “Warrior,” “Cochrane,” “Comet,” and the “Defence.” At the outbreak of war he was Sub-Lieut. on the “Comet,” a destroyer, and was posted to the “Defence” as Lieut. (E) in September, 1915. He was the holder of the Medal of the Royal Life Saving Society, which he won at the Royal Naval College, Dartmouth, in July, 1909.
- 24. Lieut. Gomersall. Lieut. William Ellis Gomersall, 22nd Manchester Regiment, eldest son of Mr. and Mrs. Hubert Gomersall, of 69, Queen’s Road, Urmston, Manchester, and nephew of the late Rev. W. J. Gomersall, was killed in action on July 1st, 1916. Lieut. Gomersall was born at Hellifield and was 21 years of age. He was educated at Manchester Grammar School and passed his Classical Matriculation in 1911. He was a member of the Grammar School O.T.C. and Lieut. in the Urmston (Manchester) Company of the Church Lads’ Brigade. He also held the certificate of the Royal Life Saving Society and St. John Ambulance Society. Soon after outbreak of war, he enlisted as a private in the Public Schools Brigade, Manchester Bn. After several months’ training at Epsom, he was selected for a commission in the 23rd Bn. Manchester Regiment, but was afterwards transferred to the 22nd Bn. After being promoted to full Lieut. on May 11th, 1915, he entered a course at the Royal Staff College, Camberley, and eventually accompanied his regiment to France where he met his death.
- 25. Major Metcalfe. Major John Chayter Metcalfe, Cheshire Regiment, killed in action July 3rd, 1916. The deceased officer, who was 34 years of age, was the only son of the late Major John A. Metcalfe, of Ings House, Hawes, and resided at King Edward’s Place, Wanborough, Wilts. He served with the 3rd West Yorkshire Regiment during the South African War, where he was for some time temporary Aide-de-Camp to the late Lord Cloughton. He afterwards received a commission in the 13th Hussars. He left the army in 1906, took up racing, and was a successful amateur steeplechase rider and breeder of thoroughbred horses. When the war commenced, he enlisted in the Public Schools Corps, and in October, 1914, received a commission in the Cheshire Regiment, being gazetted Temporary Major in September, 1915. His grandfather, who resided at Hawes, was Chaplain to the Duke of Wellington, and his uncle, the late Dr. Parker, was for years the vicar of Hawes.
- 26. Lieut. Fisher. Lieut. Henry Bryan Fisher, 5th Northumberland Fusiliers, the younger son of Dr. G. E. Fisher, of Skipton, died in hospital from severe wounds in both thighs, in July, 1916. The deceased officer, who was only 20 years of age, was educated at Colwall School and Winchester College, and afterwards went to Canada, where he was engaged in farming with his uncle, Mr. Arthur Fisher. After the outbreak of war, he returned to England to enlist, and receiving a commission was gazetted to the 5th Northumberland Fusiliers.
- 27. 2nd Lieut. Parsons. 2nd Lieut. Ernest Parsons, Duke of Wellington’s (West Riding) Regiment (attached 1st Welsh Regt.), died from fever in the Malaria Hospital, at Salonica, on July 24th, 1916. Lieut. Parsons left Chili in November, 1914, where he had an excellent railway appointment, with twenty-eight other Englishmen, who had come along with him 11,000 miles to offer their services to the country. They drilled daily on board ship, and on arrival in England enlisted early in 1915. Lieut. Parsons received his commission, and after being stationed at Woolwich, Aldershot and Basingstoke, he sailed for Egypt in November, 1915. In May, 1916, he was transferred to Salonica. Lieut. Parsons was the younger son of Mr. and Mrs. R. Parsons, School House, Rathmell, where he was born. He was 25 years of age.
- 28. 2nd Lieut. Dinsdale. 2nd Lieut. Frank Dinsdale, York Lancaster Regiment, son of Mr. Mrs. Jas. Dinsdale, Show Cote, Askrigg, was reported wounded and missing on July 1st, 1916, and afterwards officially presumed killed in action. Lieut. Dinsdale received his commission from the Leeds University O.T.C. in Sept. 1915, and went to France in June 1916.
- 29. Lieut. Fryer. 2nd Lieut. James W. Fryer, Northumberland Fusiliers, only son of the late Major J. W. Fryer and Mrs. Fryer, now of “Kingarth,” Fenham, Newcastle, and formerly of Hawes, killed in action on July 1st, 1916. He joined the Northumberland Fusiliers shortly after the outbreak of war. Prior to joining the forces he was articled clerk to Messrs. Dickinson, Millar Turnbull, Solicitors, Newcastle, and had successfully passed his first examinations, and was hoping to qualify his final when he joined the Colours. Major Maufe. Major S. Broadbent Maufe, West Yorkshire Regt., died of wounds on July 5th, 1916. The late Major Maufe was the eldest son of Frederick Broadbent and Helen Mann Maufe, of Warlbeck, Ilkley, and husband of Hilda Maufe, of Acomb, York. He was educated at Uppingham and Clare College, joined the army in 1910, and was mentioned in despatches.
- 30. Capt. Horsfall. Captain Cedric F. Horsfall, ⅙ th Duke of Wellington’s (West Riding) Regiment, eldest son of Sir John and Lady Horsfall, of Hayfield, Glusburn, was killed in action in France on September 18th, 1916. Captain Horsfall, who was a popular figure in Craven, joined the Army at the outbreak of hostilities. Prior to doing so, he was in partnership with his father at Hayfield Mills. When his regiment went to the front, he was a 2nd Lieut., and a few months’ service gained him his Captaincy, but not before he had been wounded twice. It was in May, 1916, that he returned to active service. Up to joining the forces he had taken a deep interest in politics. He was a member of the Sutton Baptist Church, and was 26 years of age.
- 31. 2nd Lieut. Goodall. 2nd Lieut. Arthur C. Goodall, Yorkshire Regiment (Green Howards), nephew of Mr. and Mrs. J. Goodall, of “New Zealand,” Crosshills, killed in action in France, November 6th, 1916. Lieut. Goodall was a member of the Durham Light Infantry (T) when hostilities broke out. Shortly after the outbreak, he was promoted to non-commissioned rank and went to the Front in April, 1915. In December, 1915, he received a well-earned commission, but in March, 1916, he was invalided home. Lieut. Goodall was well-known in the Crosshills district, where he resided for some time.
- 32. Lieut. Cutliffe Hyne. Lieut. G. C. H. Cutliffe Hyne, Irish Guards, only son of Mr. C. J. Cutliffe Hyne, of Kettlewell, the well-known Yorkshire novelist, died on November 21st, 1916, at the home of the Hon. Mrs. F. Guest, used as a Military Hospital, in Park Lane, London. The interment took place on November 25th, 1916, with military honours, at Kettlewell. Lieut. Hyne, who was in his 19th year, figured in a notable deed of gallantry, which resulted in the saving of all his guns, but seventeen men out of twenty-five in his gun company became casualties. He was wounded about the middle of September, 1916, and taken to the hospital named above.
- 33. Lieut. Snowden. Lieut. Jasper Whitfield Snowden, Worcestershire Regiment, the only son of Mr. and Mrs. Edward Snowden, of the Garth, Embsay, and grandson of the late Rev. John Snowden, Vicar of Ilkley, was born at Bradford, in 1896. The deceased officer commenced his education at the Bradford Grammar School, and in 1909 went to Rossall School. He took a keen interest in sport and the study of natural history, and gained several prizes from the Royal Society for the Protection of Birds. From the very first he was an enthusiastic and active member of the Officers’ Training Corps, and was in Camp at Tidworth when war broke out. He volunteered for service and was given a commission in the above regiment, was sent to France on February 17th, 1915, and was wounded at St. Eloi, in May. In September he was sent to the Dardanelles where, owing to an attack of dysentery, he went into hospital. He soon rejoined his Bn. in Egypt, and later was drafted to Mesopotamia where he was again wounded on April 5th, 1916. He was afterwards sent to India where he was for some months on sick leave. He returned to the Tigris Line at the end of January, and fell in action on February 25th, 1917.
- 34. Lieut. Wilson. Lieut. Alec. Wilson, 1st Herefordshire Regiment (T.F.), son of Mr. and Mrs. F. J. Wilson, J.P., of Lothersdale. Lieut. Wilson, whose stay at Ermysted’s Grammar School, Skipton, extended from 1906 to 1913, and included the honoured position of Captain, was articled in the estate office of the Marquis of Abergavenny. When war broke out he at once joined the above regiment as 2nd Lieut. He went out to the Dardanelles in August, 1915, and went through that trying and devastating campaign. Following the evacuation, Lieut. Wilson was sent to Egypt, where he was killed on March 26th, 1917.
- 35. Lieut. Whaley. Lieut. Frank Whaley, Yorkshire Regiment, the youngest son of the Rev. F. W. Whaley, Vicar of Horton-in-Ribblesdale, was killed in action on March 31st, 1917. He joined the Royal Fusiliers (Public Schools Battalion) as a private soon after the outbreak of war, and, after serving at the Front in France for six months in 1915-1916, was sent home to train for a commission, being gazetted 2nd Lieut. on 26th September, 1916. He joined his regiment in France early in November, 1916.
- 36. 2nd Lieut. Broughton. 2nd Lieut. Thomas D. Broughton, King’s Own Yorkshire Light Infantry, son of the late Mr. Thomas Broughton and Mrs. Broughton, of Park House, Highfield Terrace, Skipton, died in hospital in April, 1917, from wounds received in action. After leaving school, Lieut. Broughton attended the Technical College, Bradford, for several years, and subsequently joined the firm of Messrs. H. A. Gray Co., Valley Mills, Bradford. He joined the Artists’ Rifles in November, 1915, and after the usual training at Camberley Staff College, was given a commission in the King’s Own Yorkshire Light Infantry.
- 37. 2nd Lieut. Goodman. 2nd Lieut. Eric G. Goodman, Dorsetshire Regiment, who was killed in action in France on April 12th, 1917, was a son of Dr. and Mrs. T. H. Goodman, 39, High Street, Haverhill, Suffolk, and a relative of Nurse Beresford, of Skipton. Twenty-two years of age, Lieut. Goodman received his education at Epsom College and Skipton Grammar School, being at the latter School Captain of the Rugby and Cricket teams and sports champion in 1910. In 1913, he joined the Civil Service Rifles, and on August 4th, 1914, he volunteered for service. He went to France in March, 1915, and was twice wounded.
- 38. Lieut. Bellamy. Lieut. T. B. Bellamy, King’s Own Royal Lancaster Regiment, son of Mr. and Mrs. T. Bellamy, Chapel Hill, Skipton, died from wounds received in action in Mesopotamia on April 30th, 1917. Formerly a traveller for Mr. John Mollet, ironmonger, Bradford, Lieut. Bellamy enlisted as a private in the R.F.A. a month after the outbreak of war, and in 1915 accepted the offer of a commission, and was posted to the regiment named. After taking part in the opening operations in Mesopotamia, he went to India on sick leave, but returned to the Near East at Christmas, 1916, and, along with two other officers of an advanced guard, was the first to enter Baghdad.
- 39. Capt. Hook. Captain Cyril Hook, Manchester Regiment, who was killed in action on April 23rd, 1917, was the second son of Sergt.-Major and Mrs. Hook (late of Bangalore), and grandson of the late Sergt.-Major Hook, of Settle. He was 21 years of age, and joined the Manchester Regiment as a 2nd Lieut. He went to the Front in November, 1915, being wounded in the July advance, but returned to active service again in November, 1916. Before the outbreak of war he was with Messrs. Mather Platt, of Manchester, and a member of the Broughton Park Rugby Club.
- 40. Capt. Mackay. Captain James Bruce Mackay, West Yorkshire Regiment, was the elder son of Mr. and Mrs. Walter Mackay, 11, High Street, Skipton. Thirty- two years of age, he enlisted as a private in the Hussars during the month following the outbreak of war, and, after twelve months’ training at Colchester, accepted a commission and was gazetted 2nd Lieut., being posted to the West Yorkshire Regt., proceeding to France in July, 1916. He took part in the battle of the Somme, and his promotion to the rank of Captain came on the battlefield. He was killed in action on May 3rd, 1917. He was educated at the Skipton Grammar School, and prior to enlisting was an Inspector in the London district for Messrs. Freeman, Hardy Willis, boot manufacturers.
- 41. Lieut. Marlor. 2nd Lieut. Eric Marlor, Duke of Wellington’s (West Riding) Regiment, son of Mr. Frank Marlor, Close House, Settle. Joined the Artists’ Rifles, December 15th, 1915, as a private. He afterwards entered the Cadet School, and was gazetted 2nd Lieut. of the 2 /6th Duke of Wellington’s Regt., in September, 1916. In December of the same year he was attached to the 2 /7th Duke of Wellington’s Regt., and went to France in January, 1917. He was reported missing May 3rd, 1917, in the attack on Bullecourt, and was last seen passing through the German wire at the head of his men. Neither 2nd Lieut. Marlor nor any of his men reported missing on that day have ever been traced or heard of, and the War Office has officially presumed their deaths.
- 42. Capt. Bennett. Captain Claude D. Bennett, 2 /6th Duke of Wellington’s (West Riding) Regiment, was the son of the late Mr. John Bennett, of Airedale Terrace, Skipton. The deceased officer was wounded in May, 1917, but he made a good recovery and returned to France on Wednesday, July 11th, 1917, exactly a week before he received the wounds which resulted in his death the same day. After leaving Skipton Grammar School, Captain Bennett served an apprenticeship to the teaching profession with the late Mr. W. H. Walker, of the Water Street Council School, Skipton, and later went to Westminster College to complete his training. Afterwards, for a time, he was a teacher at the Water Street School, where he remained about twelve months. From this school he was transferred to the staff of the Old British School at Skipton, and afterwards was for four years assistant master at the Brougham Street Council School, Skipton. Subsequently he was appointed headmaster of Langcliffe Council School. Captain Bennett, who was 30 years of age, started service in the Army as 2nd Lieut., and his promotion was rapid.
- 43. 2nd Lieut. Gill. 2nd Lieut. Frank Hubert Gill, West Yorkshire Regiment, met his death in action on August 16th, 1917. He was the youngest son of Mr. and Mrs. John Gill, of Park Avenue, Skipton. Twenty-three years of age, deceased was educated at the Water Street Wesleyan Higher Grade School, and Ermysted’s Grammar School, Skipton. Before the war, he held a commission in the local Cadet Corps, but on the outbreak of hostilities, he joined the ⅙ th Duke of Wellington’s Regiment as a private, with which he served in France for twenty-three months. Subsequently he was granted a commission, and returned home in the early part of 1917 to undergo the necessary training for his new duties in a school at Fleet. Afterwards he was attached to the West Yorkshire Regiment, and again proceeded to the front about five weeks before he met his death.
- 44. 2nd Lieut. Lodge. 2nd Lieut. B. G. Lodge, Durham Light Infantry, was killed in action on August 24th, 1917. Deceased, who was 23 years of age, joined the Royal Fusiliers in December, 1914. After training at several camps, he went to France with his regiment in November, 1915, returned to England in April, 1916, and was gazetted on the 9th August, 1916, as 2nd Lieut. in the Special Reserve D.L.I. In September, 1916, he again went out to France, and after acting as Company Bombing Officer, he was Battn. Bombing Officer and Intelligence Officer until he was killed. He was educated at the Minster Yard School, York, Yorebridge Grammar School and Giggleswick School, where he was a member of the O.T.C.
- 45. Capt. M. D. W. Maude. Captain M. D. W. Maude, Yorkshire Regiment, younger son of Lt.-Col. and Mrs. Maude, The Fleets, Rylstone-in-Craven, died in a military hospital at Dover, on October 14th, 1917, from wounds received in action. Captain Maude went to France in 1914 with the seventh division, and was in the first battle of Ypres. He was mentioned in despatches after the battle of the Somme. He was 27 years of age, and was attached to the West Yorkshire Regiment (Special Reserve).
- 46. Capt. G. W. E. Maude. Capt. G. W. E. Maude, died of pneumonia at Peshawar, India, November 5th, 1919. Gerald William Edward Maude was the elder and only surviving son of Lieut.-Colonel W. W. and Mrs. Maude, The Fleets, Rylstone-in-Craven. Capt. Maude had served eight years in India with his regiment, the 1st Battalion, A.P.W.O., Yorkshire Regiment (Green Howards), and in the spring of 1919 he was badly wounded by a bullet penetrating one of his lungs near Fort Dekka in Baluchistan. After three months sick leave in Kashmir, he recovered sufficiently to resume duty. On being granted a year’s leave he was hoping to embark for England on November 7th, 1919, but unfortunately he contracted a cold, which probably affected the injured lung. Pneumonia developed, and Captain Maude died on November 5th at the Military Hospital, Peshawar. He leaves a widow and one little son, and was 30 years of age.
- 47. Lieut. Styles, M.C. 2nd Lieut. H. T. Styles, M.C., Manchester Regiment, son of Mr. and Mrs. Styles, Harden Cottage, Austwick, was killed on October 2nd, 1917, aged 23 years. 2nd Lieut. Rodwell, M.C.
- 48. 2nd Lieut. Wm. Albert Rodwell, M.C., Royal Engineers, younger son of Mr. and Mrs. A. Rodwell, of West Ville, Skipton, was killed in France on Nov. 9th, 1917. Twenty years of age, Lieut. Rodwell was educated at the Skipton Wesleyan Higher School. He was offered and accepted a commission in the army in October, 1915. He was then attached to the 20th Bn. Durham Light Infantry, and went to France in May, 1916. 2nd Lieut. Gladstone. 2nd Lieut. Ralph O. Gladstone, Royal Engineers, of Holme Road, Crosshills, was killed in action on November 2nd, 1917, in France. Shortly before the outbreak of war, he was working in Spain for the British Thomson-Houston Co., of Rugby, and, immediately on the outbreak of hostilities he joined the Royal Engineers as a private, and went to France in 1914. He was wounded at the first battle of Ypres. He was 27 years of age.
- 49. 2nd Lieut. Watson. 2nd Lieut. G. W. A. Watson, Royal Air Force, eldest son of the late Mr. James Watson, Conistone-with-Kilnsey, was killed in action on March 7th, 1918. Lieut. Watson was only 20 years of age, and when war broke out he enlisted as a seaman in the Royal Naval Division. Later he joined the Royal Air Force, and went to France only a fortnight before he died. The deceased officer, who had a genius for invention, was formerly an engineering student at Leeds University, and a new fuse and engine have been made to his designs.
- 50. 2nd Lieut. Atkinson. 2nd Lieut. Victor R. Atkinson, ⅙th Duke of Wellington’s (West Riding) Regiment, was killed in action in France on November 23rd, 1917. He was the son of Dr. and Mrs. Atkinson, of “Bowerley,” Settle, and grandson of the late Mr. Edward Atkinson, of Leeds and Harrogate. He was educated at the Giggleswick Grammar School, where he was in the O.T.C. He was 20 years of age.
- 51. Capt. Littledale. Captain Willoughby John Littledale, Oxfordshire and Buckinghamshire Light Infantry, only son of Mr. Willoughby Aston Littledale, formerly of Bolton-by-Bowland, was killed in action on March 23rd, 1918. Born in 1896, he was educated at Copthorne School and Eton, and was accepted for entrance at Trinity College, Oxford, but on the outbreak of war proceeded instead to Sandhurst, receiving his commission in December, 1914. He went to France in May, 1915, and was wounded in November, 1916. He afterwards rejoined his regiment and was killed, as stated above, when commanding his company in the front line. Flight Lieut. Brookes. Flight Lieut. R. B. Brookes, Royal Air Force, son of Mr. John Brooks, Greta Villas, Ingleton, officially presumed to have been killed on March 13th, 1918.
- 52. Major Walling, M.C., C. de G. Major E. Walling, M.C., Croix de guerre, West Yorkshire Regiment, eldest son of Mr. and Mrs. G. Walling, Ferncliffe, Ingleton, was killed in action at Kemmel Hill on April 23rd, 1918. Major Walling, who was twice Mentioned in Despatches, had a brilliant scholastic career. Obtaining a County Minor Scholarship, he went to Giggleswick Grammar School, and there passed his Matriculation, 1st division, and took a County Major Scholarship and a Natural Science Exhibition at Magdalen College, Oxford. He was at Oxford four years, and was in the hockey and football teams. He was a keen golfer and a member of the Leeds and Ingleton Clubs. Leaving college he went as master at Dulwich, Oxford High School, Sheffield Grammar School, and Leeds Grammar School, where he was Senior Science Master. Prior to the war he held a commission in the Territorial Force (Leeds Rifles), and was in camp at Scarborough when war broke out. He joined his regiment in France early in 1915, and served there until he was killed.
- 53. Lieut. Pettitt. Lieut. W. Pettitt, Loyal North Lancashire Regiment, of Settle, was killed during April, 1918. Lieut. G. Procter. Lieut. George Procter, Lancashire Fusiliers, only son of Mr. and Mrs. Thomas Procter, Greystones, Gisburn, killed in action on April 7th,
- 54. Welcome to our website – the ideal destination for book lovers and knowledge seekers. With a mission to inspire endlessly, we offer a vast collection of books, ranging from classic literary works to specialized publications, self-development books, and children's literature. Each book is a new journey of discovery, expanding knowledge and enriching the soul of the reade Our website is not just a platform for buying books, but a bridge connecting readers to the timeless values of culture and wisdom. With an elegant, user-friendly interface and an intelligent search system, we are committed to providing a quick and convenient shopping experience. Additionally, our special promotions and home delivery services ensure that you save time and fully enjoy the joy of reading. Let us accompany you on the journey of exploring knowledge and personal growth! ebookultra.com