This book presents state-of-the-art research on the distribution modulo one of sequences of integral powers of real numbers and related topics. Most of the results have never before appeared in one book and many of them were proved only during the last decade. Topics covered include the distribution modulo one of the integral powers of 3/2 and the frequency of occurrence of each digit in the decimal expansion of the square root of two. The author takes a point of view from combinatorics on words and introduces a variety of techniques, including explicit constructions of normal numbers, Schmidt's games, Riesz product measures and transcendence results. With numerous exercises, the book is ideal for graduate courses on Diophantine approximation or as an introduction to distribution modulo one for non-experts. Specialists will appreciate the inclusion of over 50 open problems and the rich and comprehensive bibliography of over 700 references.

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Distribution Modulo One and Diophantine Approximation

Quasicrystals and Geometry

In these notes, we begin by recalling aspects of the classical theory of metric Diophantine approximation; such as theorems of Khintchine, Jarnik, Duffin-Schaeffer and Gallagher. We then describe recent strengthening of various classical statements as well as recent developments in the area of Diophantine approximation on manifolds. The latter includes the well approximable, the badly approximable and the inhomogeneous aspects.

Metric Diophantine Approximation: aspects of recent work

An introduction to Diophantine Approximation. By J. W. S. Cassbls. Cambridge tracts in mathematics and mathematical physics, No. 45. Pp. 166. 22s. 6d. 1957. (Cambridge)

We study bounded remainder sets with respect to an irrational rotation of the d-dimensional torus. The subject goes back to Hecke, Ostrowski and Kesten who characterized the intervals with bounded remainder in dimension one.

Sets of bounded discrepancy for multi-dimensional irrational rotation

In 1998, Burago-Kleiner and McMullen independently proved the existence of separated nets in R which are not bi-Lipschitz equivalent (BL) to a lattice. A finer equivalence relation than BL is bounded displacement (BD). Separated nets arise naturally as return times to a section for minimal R-actions. We analyze the separated nets which arise via these constructions, focusing particularly on nets arising from linear R-actions on tori. We show that generically these nets are BL to a lattice, and for some choices of dimensions and sections, they are generically BD to a lattice. We also show the existence of such nets which are not BD to a lattice.

/pdf/equivalence-relations-on-separated-nets-arising-from-linear-jakox7e41z.pdf

Equivalence relations on separated nets arising from linear toral flows

Given a nonnegative function $${\psi\mathbb{N} \to \mathbb{R}}$$
 , let W(ψ) denote the set of real numbers x such that |nx − a|  0). A consequence of our main result is that W(ψ) is of full Lebesgue measure if there exists an $${\epsilon > 0}$$
 such that
 $$ \textstyle \sum_{n\in\mathbb{N}}\left(\frac{\psi(n)}{n}\right)^{1+\epsilon}\varphi (n)=\infty. $$
 The Duffin–Schaeffer Conjecture is the corresponding statement with $${\epsilon = 0}$$
 and represents a fundamental unsolved problem in metric number theory. Another consequence is that W(ψ) is of full Hausdorff dimension if the above sum with $${\epsilon = 0}$$
 diverges; i.e. the dimension analogue of the Duffin–Schaeffer Conjecture is true.

The Duffin–Schaeffer Conjecture with extra divergence

For any irrational cut-and-project setup, we demonstrate a natural infinite family of windows which gives rise to separated nets that are each bounded distance to a lattice. Our proof provides a new construction, using a sufficient condition of Rauzy, of an infinite family of non-trivial bounded remainder sets for any totally irrational toral rotation in any dimension.

/pdf/constructing-bounded-remainder-sets-and-cut-and-project-sets-srfwlyzm7k.pdf

Constructing bounded remainder sets and cut-and-project sets which are bounded distance to lattices

Given a nonnegative function ${\psi\mathbb{N} \to \mathbb{R}}$, let W(ψ) denote the set of real numbers x such that |nx − a| 0). A consequence of our main result is that W(ψ) is of full Lebesgue measure if there exists an ${\epsilon > 0}$ such that

$$ \textstyle \sum_{n\in\mathbb{N}}\left(\frac{\psi(n)}{n}\right)^{1+\epsilon}\varphi (n)=\infty. $$

The Duffin–Schaeffer Conjecture is the corresponding statement with ${\epsilon = 0}$ and represents a fundamental unsolved problem in metric number theory. Another consequence is that W(ψ) is of full Hausdorff dimension if the above sum with ${\epsilon = 0}$ diverges; i.e. the dimension analogue of the Duffin–Schaeffer Conjecture is true.

/pdf/the-duffin-schaeffer-conjecture-with-extra-divergence-ii-59kuix0nve.pdf

The Duffin–Schaeffer conjecture with extra divergence II

There are two main interrelated goals of this paper. Firstly we investigate the sums $S_N(\alpha,\gamma):=\sum_{n=1}^N\frac{1}{n\|n\alpha-\gamma\|}$ and $R_N(\alpha,\gamma):=\sum_{n=1}^N\frac{1}{\|n\alpha-\gamma\|},$ where $\alpha$ and $\gamma$ are real parameters and $\|\cdot\|$ is the distance to the nearest integer. Our theorems improve upon previous results of W.M.Schmidt and others, and are (up to constants) best possible. Related to the above sums, we also obtain upper and lower bounds for the cardinality of $\{1\le n\le N:\|n\alpha-\gamma\|<\ve\} \, ,$ valid for all sufficiently large $N$ and all sufficiently small $\ve$.This first strand of the work is motivated by applications to multiplicative Diophantine approximation, which are also considered. In particular, we obtain complete Khintchine type results for multiplicative simultaneous Diophantine approximation on fibers in $\R^2$. The divergence result is the first of its kind and represents an attempt of developing the concept of ubiquity to the multiplicative setting.

Alan Haynes

Papers

Equivalence relations on separated nets arising from linear toral flows

The Duffin–Schaeffer Conjecture with extra divergence

Constructing bounded remainder sets and cut-and-project sets which are bounded distance to lattices

The Duffin–Schaeffer conjecture with extra divergence II

Sums of Reciprocals of Fractional Parts and Multiplicative Diophantine Approximation