On the construction of small subsets containing special elements in a finite field

In this note we construct a series of small subsets containing a non-d-th power element in a finite field by applying certain bounds on incomplete character sums. Precisely, let $h=\lfloor q^{\delta}\rfloor>1$ and $d\mid q^h-1$. Let $r$ be a prime divisor of $q-1$ such that the largest prime power part of $q-1$ has the form $r^s$. Then there is a constant $0<\epsilon<1$ such that for a ratio at least ${q^{-\epsilon h}}$ of $\alpha\in \mathbb{F}_{q^{h}} \backslash\mathbb{F}_{q}$, the set $S=\{ \alpha-x^t, x\in\mathbb{F}_{q}\}$ of cardinality $1+\frac {q-1} {M(h)}$ contains a non-d-th power in $\mathbb{F}_{q^{\lfloor q^\delta\rfloor}}$, where $t$ is the largest power of $r$ such that $t<\sqrt{q}/h$ and $M(h)$ is defined as $$M(h)=\max_{r \mid (q-1)} r^{\min\{v_r(q-1), \lfloor\log_r{q}/2-\log_r h\rfloor\}}.$$ Here $r$ runs thourgh prime divisors and $v_r(x)$ is the $r$-adic oder of $x$. For odd $q$, the choice of $\delta=\frac 12-d, d=o(1)>0$ shows that there exists an explicit subset of cardinality $q^{1-d}=O(\log^{2+\epsilon'}(q^h))$ containing a non-quadratic element in the field $\mathbb{F}_{q^h}$. On the other hand, the choice of $h=2$ shows that for any odd prime power $q$, there is an explicit subset of cardinality $1+\frac {q-1}{M(2)}$ containing a non-quadratic element in $\mathbb{F}_{q^2}$. This improves a $q-1$ construction by Coulter and Kosick \cite{CK} since $\lfloor \log_2{(q-1)}\rfloor\leq M(2) < \sqrt{q}$. In addition, we obtain a similar construction for small sets containing a primitive element. The construction works well provided $\phi(q^h-1)$ is very small, where $\phi$ is the Euler's totient function.