arXiv subjects

Lisa Kaltenböck

Publications and source records attributed to Lisa Kaltenböck.

7 recordsLinked to original sources

arXiv2023

A positive lower bound for $\liminf_{N\to\infty} \prod_{r=1}^N \left| 2\sin πr φ\right|$

Nearly 60 years ago, Erdős and Szekeres raised the question of whether $$\liminf_{N\to \infty} \prod_{r=1}^N \left| 2\sin πr α\right| =0$$ for all irrationals $α$. Despite its simple formulation, the question has remained unanswered. It was shown by Lubinsky in 1999 that the answer is yes if $α$ has unbounded continued fraction coefficients, and it was suggested that the answer is yes in general. However, we show in this paper that for the golden ratio $φ=(\sqrt{5}-1)/2$, $$\liminf_{N\to \infty} \prod_{r=1}^N \left| 2\sin πr φ\right| >0 ,$$ providing a negative answer to this long-standing open problem.

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arXiv2020

Sums of averages of gcd-sum functions II

Let $\gcd(k,j)$ denote the greatest common divisor of the integers $k$ and $j$, and let $r$ be any fixed positive integer. Define $$ M_r(x; f) := \sum_{k\leq x}\frac{1}{k^{r+1}}\sum_{j=1}^{k}j^{r}f(\gcd(j,k)) $$ for any large real number $x\geq 5$, where $f$ is any arithmetical function. Let $ϕ$, and $ψ$ denote the Euler totient and the Dedekind function, respectively. In this paper, we refine asymptotic expansions of $M_r(x; {\rm id})$, $M_r(x;ϕ)$ and $M_r(x;ψ)$. Furthermore, under the Riemann Hypothesis and the simplicity of zeros of the Riemann zeta-function, we establish the asymptotic formula of $M_r(x;{\rm id})$ for any large positive number $x>5$ satisfying $x=[x]+\frac{1}{2}$.

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arXiv2019

Pair Correlations of Niederreiter and Halton Sequences are not Poissonian

Niederreiter and Halton sequences are two prominent classes of multi-dimensional sequences which are widely used in practice for numerical integration methods because of their excellent distribution qualities. In this paper, we show that these sequences - even though they are uniformly distributed - fail to satisfy the stronger property of Poissonian pair correlations. This extends already established results for one-dimensional sequences and confirms a conjecture of Larcher and Stockinger. The proofs rely on a general tool which identifies specific regularities of a sequence to be sufficient for not having Poissonian pair correlations.

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arXiv2019

On the asymptotic behaviour of the sine product $\prod_{r=1}^n|2\sin(πr α)|$

In this paper we review recently established results on the asymptotic behaviour of the trigonometric product $P_n(α) = \prod_{r=1}^n |2\sin πr α|$ as $n\to \infty$. We focus on irrationals $α$ whose continued fraction coefficients are bounded. Our main goal is to illustrate that when discussing the regularity of $P_n(α)$, not only the boundedness of the coefficients plays a role; also their size, as well as the structure of the continued fraction expansion of $α$, is important.

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arXiv2018

On a multi-dimensional Poissonian pair correlation concept and uniform distribution

The aim of the present article is to introduce a concept which allows to generalise the notion of Poissonian pair correlation, a second-order equidistribution property, to higher dimensions. Roughly speaking, in the one-dimensional setting, the pair correlation statistics measures the distribution of spacings between sequence elements in the unit interval at distances of order of the mean spacing $1/N$. In the $d$-dimensional case, of course, the order of the mean spacing is $1/N^{\frac{1}{d}}$, and --in our concept-- the distance of sequence elements will be measured by the supremum-norm. Additionally, we show that, in some sense, almost all sequences satisfy this new concept and we examine the link to uniform distribution. The metrical pair correlation theory is investigated and it is proven that a class of typical low-discrepancy sequences in the high-dimensional unit cube do not have Poissonian pair correlations, which fits the existing results in the one-dimensional case.

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arXiv2018

On Bounded Remainder Sets and Strongly Non-Bounded Remainder Sets for Sequences $(\{a_nα\})_{n\geq 1}$

We give some results on the existence of bounded remainder sets (BRS) for sequences of the form $(\{a_nα\})_{n\geq 1}$, where $(a_n)_{n\geq 1}$ - in most cases - is a given sequence of distinct integers. Further we introduce the concept of strongly non-bounded remainder sets (S-NBRS) and we show for a very general class of polynomial-type sequences that these sequences cannot have any S-NBRS, whereas for the sequence $(\{2^nα\})_{n \geq 1}$ every interval is an S-NBRS.

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arXiv2018

A survey on M. B. Levin's proofs for the exact lower discrepancy bounds of special sequences and point sets

The goal of this overview article is to give a tangible presentation of recent breakthrough works in discrepancy theory by M. B. Levin. These works provide proofs for the exact lower discrepancy bounds of Halton's sequence and a certain class of $(t, s)$-sequences. Our survey aims at highlighting the major ideas of the proofs and we discuss further implications of the employed methods. Moreover, we derive extensions of Levin's results.

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