We investigate the discrepancy between the distributions of the random variable $\log L (\sigma , f \times f, X)$ and that of $\log L(\sigma +it, f \times f)$, that is,

$$ \begin{align*} D_{\sigma} (T) := \sup_{\mathcal{R}} |\mathbb{P}_T(\log L(\sigma+it, f \times f) \in \mathcal{R}) - \mathbb{P}(\log L(\sigma, f \times f, X) \in \mathcal{R})|, \end{align*} $$

where the supremum is taken over rectangles $\mathcal {R}$ with sides parallel to the coordinate axes. For fixed $T>3$ and $2/3 <\sigma _0 < \sigma < 1$, we prove that

$$ \begin{align*} D_{\sigma} (T) \ll \frac{1}{(\log T)^{\sigma}}. \end{align*} $$

We derive an explicit formula for the N-point correlation $F_N(s)$ of the van der Corput sequence in base $2$ for all $N \in \mathbb {N}$ and $s \geq 0$. The formula can be evaluated without explicit knowledge about the elements of the van der Corput sequence. This constitutes the first example of an exact closed-form expression of $F_N(s)$ for all $N \in \mathbb {N}$ and all $s \geq 0$ which does not require explicit knowledge about the involved sequence. Moreover, it can be immediately read off that $\lim _{N \to \infty } F_N(s)$ exists only for $0 \leq s \leq 1/2$.

Celebrated theorems of Roth and of Matoušek and Spencer together show that the discrepancy of arithmetic progressions in the first $n$ positive integers is $\Theta (n^{1/4})$. We study the analogous problem in the $\mathbb {Z}_n$ setting. We asymptotically determine the logarithm of the discrepancy of arithmetic progressions in $\mathbb {Z}_n$ for all positive integer $n$. We further determine up to a constant factor the discrepancy of arithmetic progressions in $\mathbb {Z}_n$ for many $n$. For example, if $n=p^k$ is a prime power, then the discrepancy of arithmetic progressions in $\mathbb {Z}_n$ is $\Theta (n^{1/3+r_k/(6k)})$, where $r_k \in \{0,1,2\}$ is the remainder when $k$ is divided by $3$. This solves a problem of Hebbinghaus and Srivastav.

Given any rectangular polyhedron $3$ -manifold $\mathcal {P}$ tiled with unit cubes, we find infinitely many explicit directions related to cubic algebraic numbers such that all half-infinite geodesics in these directions are uniformly distributed in $\mathcal {P}$ .

Given an infinite subset $\mathcal{A} \subseteq\mathbb{N}$ , let A denote its smallest N elements. There is a rich and growing literature on the question of whether for typical $\alpha\in[0,1]$ , the pair correlations of the set $\alpha A (\textrm{mod}\ 1)\subset [0,1]$ are asymptotically Poissonian as N increases. We define an inhomogeneous generalisation of the concept of pair correlation, and we consider the corresponding doubly metric question. Many of the results from the usual setting carry over to this new setting. Moreover, the double metricity allows us to establish some new results whose singly metric analogues are missing from the literature.

Given a finite set $A \subseteq \mathbb{R}^d$, points $a_1,a_2,\dotsc,a_{\ell} \in A$ form an $\ell$-hole in A if they are the vertices of a convex polytope, which contains no points of A in its interior. We construct arbitrarily large point sets in general position in $\mathbb{R}^d$ having no holes of size $O(4^dd\log d)$ or more. This improves the previously known upper bound of order $d^{d+o(d)}$ due to Valtr. The basic version of our construction uses a certain type of equidistributed point sets, originating from numerical analysis, known as (t,m,s)-nets or (t,s)-sequences, yielding a bound of $2^{7d}$. The better bound is obtained using a variant of (t,m,s)-nets, obeying a relaxed equidistribution condition.

In this paper, we derive geometric and analytic properties of invariant sets, including orbit closures, of a large class of piecewise-affine maps $T$ on $\mathbb{R}^{d}$. We assume that (i) $T$ consists of finitely many affine maps defined on a Borel measurable partition of $\mathbb{R}^{d}$, (ii) there is a lattice $\mathscr{L}\subset \mathbb{R}^{d}$ that contains all of the mutual differences of the translation vectors of these affine maps, and (iii) all of the affine maps have the same linear part that is an automorphism of $\mathscr{L}$. We prove that finite-volume invariant sets of such piecewise-affine maps always consist of translational tiles relative to this lattice, up to some multiplicity. When the partition is Jordan measurable, we show that closures of bounded orbits of $T$ are invariant and yield Jordan measurable tiles, again up to some multiplicity. In the latter case, we show that compact $T$-invariant sets also consist of Jordan measurable tiles. We then utilize these results to quantify the rate of convergence of ergodic averages for $T$ in the case of bounded single tiles.

We consider finite point subsets (distributions) in compact metric spaces. In the case of general rectifiable metric spaces, non-trivial bounds for sums of distances between points of distributions and for discrepancies of distributions in metric balls are given (Theorem 1.1). We generalize Stolarsky’s invariance principle to distance-invariant spaces (Theorem 2.1). For arbitrary metric spaces, we prove a probabilistic invariance principle (Theorem 3.1). Furthermore, we construct equal-measure partitions of general rectifiable compact metric spaces into parts of small average diameter (Theorem 4.1).

An important result of Weyl states that for every sequence $(a_{n})_{n\geqslant 1}$ of distinct positive integers the sequence of fractional parts of $(a_{n}{\it\alpha})_{n\geqslant 1}$ is uniformly distributed modulo one for almost all ${\it\alpha}$. However, in general it is a very hard problem to calculate the precise order of convergence of the discrepancy of $(\{a_{n}{\it\alpha}\})_{n\geqslant 1}$ for almost all ${\it\alpha}$. In particular, it is very difficult to give sharp lower bounds for the speed of convergence. Until now this was only carried out for lacunary sequences $(a_{n})_{n\geqslant 1}$ and for some special cases such as the Kronecker sequence $(\{n{\it\alpha}\})_{n\geqslant 1}$ or the sequence $(\{n^{2}{\it\alpha}\})_{n\geqslant 1}$. In the present paper we answer the question for a large class of sequences $(a_{n})_{n\geqslant 1}$ including as a special case all polynomials $a_{n}=P(n)$ with $P\in \mathbb{Z}[x]$ of degree at least 2.

In this paper we study the pseudorandom properties of the signs of Kloosterman sums.

Let $p$ be prime and let $\vartheta \,\in \,\mathbb{Z}_{p}^{*}$ be of multiplicative order $t$ modulo $p$. We consider exponential sums of the form

$$S\left( a \right)\,=\,\sum\limits_{x=1}^{t}{\exp \left( 2\pi ia{{\vartheta }^{{{x}^{2}}}}\,/\,p \right)}$$

and prove that for any $\varepsilon \,>\,0$

$$\underset{\gcd (a,\,p)\,=\,1}{\mathop{\max }}\,\,\left| S\left( a \right) \right|\,=\,O\left( {{t}^{5/6+\varepsilon }}\,{{p}^{1/8}} \right)$$

We study the Lw-norm (2 ≤ W < ∞) of the discrepancy of a sequence of points in the unit cube relative to similar copies of a given convex polygon. In particular, we prove the rather surprising result that the estimates obtained have the same order of magnitude as the analogous question when the sequence of points is replaced by a set of points.