Blocking Sets and Power Residue Modulo Integers with Bounded Number of Prime Factors

TL;DR

This paper establishes a connection between blocking sets in projective geometry and the distribution of $q$th power residues modulo integers with limited prime factors, providing bounds and classifications for such sets.

Contribution

It introduces a geometric characterization of sets avoiding perfect $q$th powers, linking number theory with Galois geometry to analyze their properties.

Findings

Sets avoiding perfect $q$th powers relate to $k$-blocking sets in projective geometry.

Lower bounds for the size of such sets are derived using geometric methods.

Complete classifications of minimal such sets are provided.

Abstract

Let $q$ be an odd prime and $k$ be a natural number. We show that a finite subset of integers $S$ that does not contain any perfect $q^{th}$ power, contains a $q^{th}$ power residue modulo almost every natural numbers $N$ with at most $k$ prime factors if and only if $S$ corresponds to a $k$-blocking set of $\PG(\mathbb{F}_{q}^{n})$. Here, $n$ is the number of distinct primes that divides the $q$-free parts of elements of $S$. Consequently, this geometric connection enables us to utilize methods from Galois geometry to derive lower bounds for the cardinalities of such sets $S$ and to completely characterize such $S$ of the smallest and the second smallest cardinalities. Furthermore, the property of whether a finite subset of integers contains a $q^{th}$ power residue modulo almost every integer $N$ with at most $k$ prime factors is invariant under the action of projective general linear group $\mathrm{PGL}(n, q)$.