关于有限域中含有特殊元素的小子集的构造
On the construction of small subsets containing special elements in a finite field
摘要：
本文通过对不完全特征和应用一定的界，构造了有限域中一系列包含非d次幂元的小子集。准确地说，让$h=\lfloor q^δ\rfloor>1$和$d\mid q^h-1$。设$r$是$q-1$的一个素因子，使得$q-1$的最大素幂部分具有$r^s$的形式。然后有一个常数$0<ε<1$，使得对于至少$α\in \mathbb{F}_{q^{h}} \backslash\mathbb{F}_{q}$的$ {q^{-εh}}$的比率，基数$1+\frac {q-1} {M(h)}$的集合$S=\{ α-x^t, x\in\mathbb{F}_{q}\}$包含$\mathbb{F}_{q^{\lfloor q^δ\rfloor}}$中的非d次幂，其中$t$是$r$的最大幂，使得$t<\sqrt{q}/h$和$M(h)$被定义为$$M(h)=\max_{r \mid (q-1)} r^{\min\{v_r(q-1), \lfloor\log_r{q}/2-\log_r h\rfloor\}}.$$这里$r$通过素数因子运行，$v_r(x)$是$x$的$r$-adic阶。对于奇数$q$，$δ=\frac 12-d, d=o(1)>0$的选择表明在域$\mathbb{F}_{q^h}$中存在包含非二次元素的基数$q^{1-d}=O(\log^{2+ε'}(q^h))$的显式子集。另一方面，$h=2$的选择表明，对于任何奇素数幂$q$，存在包含$\mathbb{F}_{q^2}$中的非二次元素的基数$1+\frac {q-1}{M(2)}$的显式子集。这改进了Coulter和Kosick \cite{CK}自$\lfloor \log_2{(q-1)}\rfloor\leq M(2) < \sqrt{q}$以来的$q-1$构造。此外，我们得到了一个类似的结构包含一个原始元素的小集合。如果$ϕ$非常小，则该构造工作良好，其中$ϕ$是欧拉函数。
英文摘要：
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 $h=\lfloor q^δ\rfloor>1$ 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 $0<ε<1$ such that for a ratio at least $ {q^{-εh}}$ of $α\in \mathbb{F}_{q^{h}} \backslash\mathbb{F}_{q}$, the set $S=\{ α-x^t, x\in\mathbb{F}_{q}\}$ of cardinality $1+\frac {q-1} {M(h)}$ contains a non-d-th power in $\mathbb{F}_{q^{\lfloor q^δ\rfloor}}$, 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 $δ=\frac 12-d, d=o(1)>0$ shows that there exists an explicit subset of cardinality $q^{1-d}=O(\log^{2+ε'}(q^h))$ 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 $ϕ(q^h-1)$ is very small, where $ϕ$ is the Euler's totient function.
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