Partitions into Piatetski-Shapiro sequences

TL;DR

This paper derives asymptotic formulas for the number of partitions of integers into Piatetski-Shapiro sequence parts, extending classical partition theory with new analytical techniques.

Contribution

It introduces asymptotic formulas for partitions into Piatetski-Shapiro sequences, utilizing Roth-Szekeres framework and equidistribution results, advancing partition analysis.

Findings

Asymptotic formulas for $p_{\kappa,m}(n)$ are established.

Framework combines Roth-Szekeres asymptotics with equidistribution techniques.

Results generalize classical partition asymptotics to Piatetski-Shapiro sequences.

Abstract

Let $\kappa$ be a positive real number and $m\in\mathbb{N}\cup\{\infty\}$ be given. Let $p_{\kappa, m}(n)$ denote the number of partitions of $n$ into the parts from the Piatestki-Shapiro sequence $(\lfloor \ell^{\kappa}\rfloor)_{\ell\in \mathbb{N}}$ with at most $m$ times (repetition allowed). In this paper we establish asymptotic formulas of Hardy-Ramanujan type for $p_{\kappa, m}(n)$, by employing a framework of asymptotics of partitions established by Roth-Szekeres in 1953, as well as some results on equidistribution.