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

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 $\{\pm p_1^{e_1}{p}_2^{e_2}\cdots p_N^{e_N}:0\le e_1,e_2,\dots ,e_N\le N\}$ 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 $m_{q,k}$.