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

Bhawesh Mishra

TL;DR

The paper proves that for any prime $q$ and any natural number $k$, the proportion of $k$-element subsets of the integers that contain a $q^{th}$ power modulo almost every prime is zero, in both additive and multiplicative senses. It introduces generalized densities on collections of subsets and reduces the problem to counting exceptional polynomials via a framework involving $q$-free parts and linear hyperplane coverings. The authors provide elementary proofs for the cases $q=2$ and odd primes, yielding explicit bounds that decay as required, and thereby establish zero additive and multiplicative subset densities for all $k$. The results strengthen the understanding of how residue properties modulo primes constrain the structure of integer subsets and relate to the theory of exceptional polynomials and Galois-theoretic criteria.

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 $m_{q,k}$.

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

TL;DR

The paper proves that for any prime and any natural number , the proportion of -element subsets of the integers that contain a power modulo almost every prime is zero, in both additive and multiplicative senses. It introduces generalized densities on collections of subsets and reduces the problem to counting exceptional polynomials via a framework involving -free parts and linear hyperplane coverings. The authors provide elementary proofs for the cases and odd primes, yielding explicit bounds that decay as required, and thereby establish zero additive and multiplicative subset densities for all . The results strengthen the understanding of how residue properties modulo primes constrain the structure of integer subsets and relate to the theory of exceptional polynomials and Galois-theoretic criteria.

Abstract

Let be a prime. We give an elementary proof of the fact that for any , the proportion of -element subsets of that contain a 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 -element subsets of that contain a power modulo almost every prime is no larger than , for some positive constant . Furthermore, the number of -element subsets of that contain a power modulo almost every prime is no larger than for some positive constant .
Paper Structure (13 sections, 6 theorems, 51 equations)

This paper contains 13 sections, 6 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Additive and Multiplicative Density of Subsets
  3. Motivation
  4. Additive and Multiplicative Subset Density
  5. Some Preliminary Results
  6. Sets that contain a $q^{th}$ power modulo almost every prime
  7. Linear Hyperplanes Associated with a Finite Subset of Integers
  8. Proof of Theorem \ref{['MainResult']} for $q = 2$
  9. Proof that $\mathcal{S}_{2,k}$ has additive subset density zero.
  10. Proof that $\mathcal{S}_{2,k}$ has multiplicative subset density zero.
  11. Proof of Theorem \ref{['MainResult']} when $q$ is an odd prime
  12. Proof that $D(\mathcal{S}_{q,k}) = 0$
  13. Proof that $D^{\times}(\mathcal{S}_{q,k}) = 0$

Key Result

Theorem 1.1

Let $q$ be a prime, $k$ be a natural number and $\mathcal{S}_{q,k}$ be the collection of $k$-element subsets of $\mathbb{Z}$ that contain a $q^{th}$ power modulo almost every prime. Then, the following hold:

Theorems & Definitions (9)

  • Theorem 1.1
  • Proposition 2.1
  • Proposition 2.2
  • Corollary 2.3
  • Remark 1
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof