Consecutive non-square non-primitive tuples in finite fields

Abstract

Let $q$ be an odd prime power and put \[ θ_q=\frac{φ(q-1)}{q-1}. \] An element of $\Fq$ is called non-square non-primitive, or \emph{NSNP}, if it is both a non-square and a non-primitive element. Let $\ell$ be an odd prime divisor of $q-1$. We obtain a character-sum estimate for the number of translates of an arbitrary finite set lying in the set of non-square $\ell$th powers. Combining this estimate with a finite computation, we prove that $θ_q<4/15$ guarantees three consecutive NSNP elements. On the boundary $θ_q=4/15$, the only exceptions are $q\in\{31,61,121\}$. As a further application, we show that for $\operatorname{char}\Fq>3$, the inequality $θ_q<8/35$ guarantees four consecutive NSNP elements, with $q=211$ as the unique exception on the boundary $θ_q=8/35$. Thus both constants are best possible.