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

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 qᵗʰ power, contains a qᵗʰ 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 (Fqⁿ). 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 qᵗʰ power residue modulo almost every integer N with at most k prime factors is invariant under the action of projective general linear group PGL(n, q).