On permutations of Hardy-Littlewood-P\'olya sequences
Christoph Aistleitner, Istvan Berkes, Robert Tichy

TL;DR
This paper investigates the asymptotic behavior of permuted sums of sequences generated by coprime integers, revealing a strong independence property that allows precise analysis of their growth depending on permutations.
Contribution
It demonstrates that these sequences satisfy a strong interlaced mixing property, enabling detailed understanding of permuted sum behaviors and their relation to Diophantine approximation.
Findings
Sequences exhibit interlaced mixing, akin to independence.
Permuted sums follow growth patterns described by Gál functions.
Behavior depends on permutation, linking to Diophantine approximation.
Abstract
Let be a finite set of coprime integers and let denote the multiplicative semigroup generated by and arranged in increasing order. The distribution of such sequences has been studied intensively in number theory and they have remarkable probabilistic and ergodic properties. For example, the asymptotic properties of the sequence are very similar to those of independent, identically distributed random variables; here denotes fractional part. However, the behavior of this sequence depends sensitively on the generating elements of and the combination of probabilistic and number-theoretic effects results in a unique, highly interesting asymptotic behavior. In particular, the properties of are not permutation invariant, in contrast to i.i.d. behavior. The purpose of this paper is to show that…
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Taxonomy
TopicsAnalytic Number Theory Research · Coding theory and cryptography · semigroups and automata theory
