Title: On the construction of small subsets containing special elements in a finite field,
Abstract: 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.