论文标题
关于有限场的Rudin-Shapiro函数的分布
On the distribution of the Rudin-Shapiro function for finite fields
论文作者
论文摘要
令$ q = p^r $为prime $ p $和$（β_1，\ ldots，β_r）$的功率，是$ \ mathbb {f} _q $ over $ \ mathbb {f} _p $的订购基础。对于$$ξ= \ sum \ limits_ {j = 1}^rx_jβ_j\ in \ mathbb {f} _q \ quad \ mbox {带digits} x_j \ in \ mathbb {f} _p，$ rudin-shapiro $ rudin-shapiro $ rud $ rudin-shapiro $ r $ r（ξ）= \ sum \ limits _ {i = 1}^{r-1} x_ix_ {i+1}，\ quad配给\ quadξ\ in \ mathbb {f} _q。 $$对于非恒定多项式$ f（x）\ in \ mathbb {f} _q [x] $和$ c \ in \ mathbb {f} _p $我们研究解决方案$ξ\ in \ mathbb {f}如果$ f（x）$的$ d $ $ d $是固定的，则$ r \ ge 6 $和$ p \ rightarrow \ infty $，对于任何$ c $，解决方案的数量均非$ p^{r-1} $。证明基于Hooley-Katz定理。
Let $q=p^r$ be the power of a prime $p$ and $(β_1,\ldots ,β_r)$ be an ordered basis of $\mathbb{F}_q$ over $\mathbb{F}_p$. For $$ ξ=\sum\limits_{j=1}^r x_jβ_j\in \mathbb{F}_q \quad \mbox{with digits }x_j\in\mathbb{F}_p, $$ we define the Rudin-Shapiro function $R$ on $\mathbb{F}_q$ by $$ R(ξ)=\sum\limits_{i=1}^{r-1} x_ix_{i+1}, \quad ξ\in \mathbb{F}_q. $$ For a non-constant polynomial $f(X)\in \mathbb{F}_q[X]$ and $c\in \mathbb{F}_p$ we study the number of solutions $ξ\in \mathbb{F}_q$ of $R(f(ξ))=c$. If the degree $d$ of $f(X)$ is fixed, $r\ge 6$ and $p\rightarrow \infty$, the number of solutions is asymptotically $p^{r-1}$ for any $c$. The proof is based on the Hooley-Katz Theorem.
