Searcharxiv⌕ Search

arXiv subjects

Christoph Aistleitner

Publications and source records attributed to Christoph Aistleitner.

At least 55 records · Page 3Linked to original sources
arXiv2016

On Weyl products and uniform distribution modulo one

In the present paper we study the asymptotic behavior of trigonometric products of the form $\prod_{k=1}^N 2 \sin(πx_k)$ for $N \to \infty$, where the numbers $ω=(x_k)_{k=1}^N$ are evenly distributed in the unit interval $[0,1]$. The main result are matching lower and upper bounds for such products in terms of the star-discrepancy of the underlying points $ω$, thereby improving earlier results obtained by Hlawka in 1969. Furthermore, we consider the special cases when the points $ω$ are the initial segment of a Kronecker or van der Corput sequence. The paper concludes with some probabilistic analogues.

math.NT↗
arXiv2016

Additive Energy and Irregularities of Distribution

We consider strictly increasing sequences $\left(a_{n}\right)_{n \geq 1}$ of integers and sequences of fractional parts $\left(\left\{a_{n} α\right\}\right)_{n \geq 1}$ where $α\in \mathbb{R}$. We show that a small additive energy of $\left(a_{n}\right)_{n \geq 1}$ implies that for almost all $α$ the sequence $\left(\left\{a_{n} α\right\}\right)_{n \geq 1}$ has large discrepancy. We prove a general result, provide various examples, and show that the converse assertion is not necessarily true.

math.NT↗
arXiv2016

On functions of bounded variation

The recently introduced concept of $\mathcal{D}$-variation unifies previous concepts of variation of multivariate functions. In this paper, we give an affirmative answer to the open question from Pausinger \& Svane (J. Complexity, 2014) whether every function of bounded Hardy--Krause variation is Borel measurable and has bounded $\mathcal{D}$-variation. Moreover, we show that the space of functions of bounded $\mathcal{D}$-variation can be turned into a commutative Banach algebra.

math.CA↗
arXiv2016

Large values of L-functions from the Selberg class

In the present paper we prove lower bounds for L-functions from the Selberg class, by this means improving earlier results obtained by the second author together with Jörn Steuding. We formulate two theorems which use slightly different technical assumptions, and give two totally different proofs. The first proof uses the "resonance method", which was introduced by Soundararajan, while the second proof uses methods from Diophantine approximation which resemble those used by Montgomery. Interestingly, both methods lead to roughly the same lower bounds, which fall short of those known for the Riemann zeta function and seem to be difficult to be improved. Additionally to these results, we also prove upper bounds for L-functions in the Selberg class and present a further application of a theorem of Chen which is used in the Diophantine approximation method mentioned above.

math.NT↗
arXiv2016

Metric results on the discrepancy of sequences $\left(a_{n} α\right)_{n \geq 1}$ modulo one for integer sequences $\left(a_{n}\right)_{n \geq 1}$ of polynomial growth

An important result of H. Weyl states that for every sequence $\left(a_{n}\right)_{n \geq 1}$ of distinct positive integers the sequence of fractional parts of $\left(a_{n} α\right)_{n\geq 1}$ is uniformly distributed modulo one for almost all $α$. However, in general it is a very hard problem to calculate the precise order of convergence of the discrepancy of $\left(\left\{a_{n} α\right\}\right)_{n \geq 1}$ for almost all $α$. 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 $\left(a_{n}\right)_{n \geq 1}$ and for some special cases such as the Kronecker sequence $\left(\left\{n α\right\}\right)_{n \geq 1}$ or the sequence $\left(\left\{n^2 α\right\}\right)_{n \geq1}$. In the present paper we answer the question for a large class of sequences $\left(a_{n}\right)_{n \geq 1}$ including as a special case all polynomials $a_{n} = P\left(n\right)$ with $P \in \mathbb{Z} \left[x\right]$ of degree at least 2.

math.NT↗
arXiv2015

Lower bounds for the maximum of the Riemann zeta function along vertical lines

Let $α\in (1/2,1)$ be fixed. We prove that $$ \max_{0 \leq t \leq T} |ζ(α+it)| \geq \exp\left(\frac{c_α(\log T)^{1-α}}{(\log \log T)^α}\right) $$ for all sufficiently large $T$, where we can choose $c_α= 0.18 (2α-1)^{1-α}$. The same result has already been obtained by Montgomery, with a smaller value for $c_α$. However, our proof, which uses a modified version of Soundararajan's "resonance method" together with ideas of Hilberdink, is completely different from Montgomery's. This new proof also allows us to obtain lower bounds for the measure of those $t \in [0,T]$ for which $|ζ(α+it)|$ is of the order mentioned above.

math.NT↗
arXiv2015

On sequences with prescribed metric discrepancy behavior

An important result of H. Weyl states that for every sequence $\left(a_{n}\right)_{n\geq 1}$ of distinct positive integers the sequence of fractional parts of $\left(a_{n} α\right)_{n \geq1}$ is uniformly distributed modulo one for almost all $α$. However, in general it is a very hard problem to calculate the precise order of convergence of the discrepancy $D_{N}$ of $\left(\left\{a_{n} α\right\}\right)_{n \geq 1}$ for almost all $α$. By a result of R. C. Baker this discrepancy always satisfies $N D_{N} = \mathcal{O} \left(N^{\frac{1}{2}+\varepsilon}\right)$ for almost all $α$ and all $\varepsilon >0$. In the present note for arbitrary $γ\in \left(0, \frac{1}{2}\right]$ we construct a sequence $\left(a_{n}\right)_{n \geq 1}$ such that for almost all $α$ we have $ND_{N} = \mathcal{O} \left(N^γ\right)$ and $ND_{N} = Ω\left(N^{γ-\varepsilon}\right)$ for all $\varepsilon > 0$, thereby proving that any prescribed metric discrepancy behavior within the admissible range can actually be realized.

math.NT↗
arXiv2015

On parametric Thue-Morse Sequences and Lacunary Trigonometric Products

One of the fundamental theorems of uniform distribution theory states that the fractional parts of the sequence $(n α)_{n \geq 1}$ are uniformly distributed modulo one (u.d. mod 1) for every irrational number $α$. Another important result of Weyl states that for every sequence $(n_k)_{k \geq 1}$ of distinct positive integers the sequence of fractional parts of $(n_k α)_{k \geq 1}$ is u.d. mod 1 for almost all $α$. However, in this general case it is usually extremely difficult to classify those $α$ for which uniform distribution occurs, and to measure the speed of convergence of the empirical distribution of $(\{n_1 α\}, ..., \{n_N α\})$ towards the uniform distribution. In the present paper we investigate this problem in the case when $(n_k)_{k \geq 1}$ is the Thue--Morse sequence of integers, which means the sequence of positive integers having an even sum of digits in base 2. In particular we utilize a connection with lacunary trigonometric products $\prod^{L}_{\ell=0} |\sin π2^{\ell} α|$, and by giving sharp metric estimates for such products we derive sharp metric estimates for exponential sums of $(n_{k} α)_{k \geq 1}$ and for the discrepancy of $(\{n_{k} α\})_{k \geq 1}.$ Furthermore, we comment on the connection between our results and an open problem in the metric theory of Diophantine approximation, and we provide some explicit examples of numbers $α$ for which we can give estimates for the discrepancy of $(\{n_{k} α\})_{k \geq1}$.

math.NT↗
arXiv2014

Convergence of series of dilated functions and spectral norms of GCD matrices

We establish a connection between the $L^2$ norm of sums of dilated functions whose $j$th Fourier coefficients are $\mathcal{O}(j^{-α})$ for some $α\in (1/2,1)$, and the spectral norms of certain greatest common divisor (GCD) matrices. Utilizing recent bounds for these spectral norms, we obtain sharp conditions for the convergence in $L^2$ and for the almost everywhere convergence of series of dilated functions.

math.CA↗
arXiv2014

Functions of bounded variation, signed measures, and a general Koksma-Hlawka inequality

In this paper we prove a correspondence principle between multivariate functions of bounded variation in the sense of Hardy and Krause and signed measures of finite total variation, which allows us to obtain a simple proof of a generalized Koksma--Hlawka inequality for non-uniform measures. Applications of this inequality to importance sampling in Quasi-Monte Carlo integration and tractability theory are given. Furthermore, we discuss the problem of transforming a low-discrepancy sequence with respect to the uniform measure into a sequence with low discrepancy with respect to a general measure $μ$, and show the limitations of a method suggested by Chelson.

math.CA↗
arXiv2014

Extremal discrepancy behavior of lacunary sequences

In 1975 Walter Philipp proved the law of the iterated logarithm (LIL) for the discrepancy of lacunary sequences: for any sequence $(n_k)_{k \geq 1}$ satisfying the Hadamard gap condition $n_{k+1} / n_k \geq q > 1,~k \geq 1,$ we have $$ \frac{1}{4 \sqrt{2}} \leq \limsup_{N \to \infty} \frac{N D_N(\{ n_1 x \}, \dots, \{n_N x\})}{\sqrt{2 N \log \log N}} \leq C_q $$ for almost all $x$. In recent years there has been significant progress concerning the precise value of the limsup in this LIL for special sequences $(n_k)_{k \geq 1}$ having a ``simple'' number-theoretic structure. However, since the publication of Philipp's paper there has been no progress concerning the lower bound in this LIL for generic lacunary sequences $(n_k)_{k \geq 1}$. The purpose of the present paper is to collect known results concerning this problem, to investigate what the optimal value in the lower bound could be, and for which special sequences $(n_k)_{k \geq 1}$ a small value of the limsup in this LIL can be obtained. We formulate three open problems, which could serve as the main targets for future research.

math.NT↗
arXiv2014

On the limit distribution of the normality measure of random binary sequences

We prove the existence of a limit distribution for the normalized normality measure $\mathcal{N}(E_N)/\sqrt{N}$ (as $N \to \infty$) for random binary sequences $E_N$, by this means confirming a conjecture of Alon, Kohayakawa, Mauduit, Moreira and R{ö}dl. The key point of the proof is to approximate the distribution of the normality measure by the exiting probabilities of a multidimensional Wiener process from a certain polytope.

math.CO↗
arXiv2014

On the law of the iterated logarithm for trigonometric series with bounded gaps II

It is well-known that for a quickly increasing sequence $(n_k)_{k \geq 1}$ the functions $(\cos 2 πn_k x)_{k \geq 1}$ show a behavior which is typical for sequences of independent random variables. If the growth condition on $(n_k)_{k \geq 1}$ is relaxed then this almost-independent behavior generally fails. Still, probabilistic constructions show that for \emph{some} very slowly increasing sequences $(n_k)_{k \geq 1}$ this almost-independence property is preserved. For example, there exists $(n_k)_{k \geq 1}$ having bounded gaps such that the normalized sums $\sum \cos 2 πn_k x$ satisfy the central limit theorem (CLT). However, due to a ``loss of mass'' phenomenon the variance in the CLT for a sequence with bounded gaps is always smaller than $1/2$. In the case of the law of the iterated logarithm (LIL) the situation is different; as we proved in an earlier paper, there exists $(n_k)_{k \geq 1}$ with bounded gaps such that $$ \limsup_{N \to \infty} \frac{\left| \sum_{k=1}^N \cos 2 πn_k x \right|}{\sqrt{N \log \log N}} = \infty \qquad \textrm{for almost all $x$.} $$ In the present paper we prove a complementary results showing that any prescribed limsup-behavior in the LIL is possible for sequences with bounded gaps. More precisely, we show that for any real number $Λ\geq 0$ there exists a sequence of integers $(n_k)_{k \geq 1}$ satisfying $n_{k+1} - n_{k} \in \{1,2\}$ such that the limsup in the LIL equals $Λ$ for almost all $x$. Similar results are proved for sums $\sum f(n_k x)$ and for the discrepancy of $(\langle n_k x \rangle)_{k \geq 1}$.

math.NT↗
arXiv2014

On the system $f(nx)$ and probabilistic number theory

Let $f: {\mathbb R}\to {\mathbb R}$ be a measurable function satisfying \begin{equation*} f(x+1)=f(x), \qquad \int_0^1 f(x)\, dx=0, \qquad \int_0^1 f^2(x)\, dx<\infty. \end{equation*} The asymptotic properties of series $\sum c_k f(kx)$ have been studied extensively in the literature and turned out to be, in general, quite different from those of the trigonometric system. As the theory shows, the behavior of such series is determined by a combination of analytic, probabilistic and number theoretic effects, resulting in highly interesting phenomena not encountered in classical harmonic analysis. In this paper we survey some recent results in the field and prove asymptotic results for the system $\{f(nx), n\ge 1\}$ in the case when the function $f$ is not square integrable.

math.NT↗
arXiv2014

On the law of the iterated logarithm for permuted lacunary sequences

It is known that for any smooth periodic function $f$ the sequence $(f(2^kx))_{k\ge 1}$ behaves like a sequence of i.i.d.\ random variables, for example, it satisfies the central limit theorem and the law of the iterated logarithm. Recently Fukuyama showed that permuting $(f(2^kx))_{k\ge 1}$ can ruin the validity of the law of the iterated logarithm, a very surprising result. In this paper we present an optimal condition on $(n_k)_{k\ge 1}$, formulated in terms of the number of solutions of certain Diophantine equations, which ensures the validity of the law of the iterated logarithm for any permutation of the sequence $(f(n_k x))_{k \geq 1}$. A similar result is proved for the discrepancy of the sequence $(\{n_k x\})_{k \geq 1}$, where $\{ \cdot \}$ denotes fractional part.

math.NT↗
arXiv2014

On permutations of lacunary series

It is a well known fact that for periodic measurable $f$ and rapidly increasing $(n_k)_{k \geq 1}$ the sequence $(f(n_kx))_{k\ge 1}$ behaves like a sequence of independent, identically distributed random variables. For example, if $f$ is a periodic Lipschitz function, then $(f(2^kx))_{k\ge 1}$ satisfies the central limit theorem, the law of the iterated logarithm and several further limit theorems for i.i.d.\ random variables. Since an i.i.d.\ sequence remains i.i.d.\ after any permutation of its terms, it is natural to expect that the asymptotic properties of lacunary series are also permutation-invariant. Recently, however, Fukuyama (2009) showed that a rearrangement of the sequence $(f(2^kx))_{k\ge 1}$ can change substantially its asymptotic behavior, a very surprising result. The purpose of the present paper is to investigate this interesting phenomenon in detail and to give necessary and sufficient criteria for the permutation-invariance of the CLT and LIL for $f(n_kx)$.

math.NT↗