Sárközy's Theorem for Fractional Monomials

Abstract

Suppose $A$ is a subset of $\{1,\dots ,N\}$ which does not contain any configurations of the form $x,x+\lfloor n^c\rfloor$ where $n\ne 0$ and $1<c<\frac{6}{5}$. We show that the density of $A$ relative to the first $N$ integers is $O_c(N^{1-\frac{6}{5c}})$. More generally, given a smooth and regular real valued function $h$ with "growth rate" $c\in (1,\frac{6}{5})$, we show that if $A$ lacks configurations of the form $x,x\pm \lfloor h(n)\rfloor$ then $\frac{|A|}{N}{\ll }_{h,\epsilon }{N}^{1-\frac{6}{5c}+\epsilon }$ for any $\epsilon >0$.

13 pages; typos corrected; the results of this paper improve on a bound proved by Sárközy and Rivat for a certain range of $c$, this is discussed in subsection 1.3