Thin subbases of Piatetski-Shapiro sequences

TL;DR

This paper investigates the structure of Piatetski-Shapiro sequences, demonstrating the existence of thin subbases of various orders and establishing asymptotic representations for their representations counts.

Contribution

It introduces new results on thin subbases within Piatetski-Shapiro sequences for different ranges of c, extending understanding of their additive properties.

Findings

Sequences contain thin subbases of specified orders for certain c ranges.

Existence of subsets with prescribed representation functions within these sequences.

Analogous results are shown for Piatetski-Shapiro primes and powers.

Abstract

For a non-integral real number $c>1$, let $\mathbb{N}_{(c)}:=\{\lfloor n^c\rfloor ~|~ n\in\mathbb{N}\}$. We show that $\mathbb{N}_{(c)}$ contains thin subbases of every order $h\geq 5$ when $1<c<2$, and $h\geq (\lfloor 2c\rfloor+1)(\lfloor 2c\rfloor+2)+1$ when $c>2$. In fact, for every regularly varying function $F$ such that \[ \frac{F(x)}{\log x}\to\infty\quad\text{ and } \quad F(x)\leq (1+o(1))\frac{\Gamma(1+1/c)^h}{\Gamma(h/c)} x^{h/c-1}, \] there exists $A\subseteq\mathbb{N}_{(c)}$ with $r_{A,h}(n)\sim F(n)$. We also establish analogous results for $k$-th powers of Piatetski-Shapiro numbers and Piatetski-Shapiro primes for small $c$.