S\'ark\"ozy's Theorem for Fractional Monomials

Abstract

Suppose $A$ is a subset of $\{1, \dotsc, N\}$ which does not contain any configurations of the form $x,x+\lfloor n^c \rfloor$ where $n \neq 0$ 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 $\frac{|A|}{N} \ll_{h,\varepsilon} N^{1-\frac{6}{5c}+\varepsilon}$ for any $\varepsilon>0$.