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

TL;DR

This paper extends Sárközy's theorem to fractional monomials, establishing upper bounds on the density of subsets avoiding specific fractional polynomial configurations, with implications for additive combinatorics.

Contribution

It generalizes Sárközy's theorem to fractional powers and smooth functions, providing new density bounds for sets avoiding fractional polynomial differences.

Findings

Sets avoiding fractional polynomial configurations have density bounds depending on the growth rate c.

The results apply to smooth, regular functions with growth rate c in (1, 6/5).

Provides quantitative bounds on the size of such sets.

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 $\varepsilon>0$.