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 nond-th power element in a finite field by applying certain bounds on incomplete character sums. Precisely, let h = ⌊qδ⌋ > 1 and d | qh − 1. Let r be a prime divisor of q− 1 such that the largest prime power part of q − 1 has the form rs. Then there is a constant 0 < ǫ < 1 such that for a ratio at least q−ǫh of α ∈ Fqh\Fq, the set S = {α − xt, x ∈ Fq} of cardinality 1 + q−1 M(h) contains a non-d-th power in F q⌊q⌋ , where t is the largest power of r such that t < √ q/h and M(h) is defined as M(h) = max r|(q−1) rrr q/2−logr . Here r runs thourgh prime divisors and vr(x) is the r-adic oder of x. For odd q, the choice of δ = 1 2 − d, d = o(1) > 0 shows that there exists an explicit subset of cardinality q1−d = O(log ′ (qh)) containing a non-quadratic element in the field Fqh . 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+ q−1 M(2) containing a non-quadratic element in Fq2 . This improves a q − 1 construction by Coulter and Kosick [6] since ⌊log2 (q − 1)⌋ ≤ M(2) < √ q. In addition, we obtain a similar construction for small sets containing a primitive element. The construction works well provided φ(qh − 1) is very small, where φ is the Euler’s totient function.