Almost No Finite Subset of Integers Contains a $q^{th}$ Power Modulo Almost Every Prime

TL;DR

This paper proves that the proportion of finite integer subsets containing a $q$th power modulo almost every prime is negligible, providing elementary bounds and showing such subsets are extremely rare.

Contribution

It offers an elementary proof that almost no finite subset of integers contains a $q$th power modulo almost every prime, with explicit bounds on their counts.

Findings

The proportion of such subsets tends to zero as N grows.

Explicit bounds are provided for subsets within intervals and specific multiplicative structures.

The results hold regardless of additive or multiplicative measure.

Abstract

Let $q$ be a prime. We give an elementary proof of the fact that for any $k\in\mathbb{N}$, the proportion of $k$-element subsets of $\mathbb{Z}$ that contain a $q^{th}$ power modulo almost every prime, is zero. This result holds regardless of whether the proportion is measured additively or multiplicatively. More specifically, the number of $k$-element subsets of $[-N, N]\cap\mathbb{Z}$ that contain a $q^{th}$ power modulo almost every prime is no larger than $a_{q,k} N^{k-(1-\frac{1}{q})}$, for some positive constant $a_{q,k}$. Furthermore, the number of $k$-element subsets of $\{\pm p_{1}^{e_{1}} p_{2}^{e_{2}} \cdots p_{N}^{e_{N}} : 0 \leq e_{1}, e_{2}, \ldots, e_{N}\leq N\}$ that contain a $q^{th}$ power modulo almost every prime is no larger than $m_{q,k} \frac{N^{Nk}}{q^{N}}$ for some positive constant $m_{q,k}$.