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- W4287764327 abstract "Let $q=p^r$ be the power of a prime $p$ and $(beta_1,ldots ,beta_r)$ be an ordered basis of $mathbb{F}_q$ over $mathbb{F}_p$. For $$ xi=sumlimits_{j=1}^r x_jbeta_jin mathbb{F}_q quad mbox{with digits }x_jinmathbb{F}_p, $$ we define the Rudin-Shapiro function $R$ on $mathbb{F}_q$ by $$ R(xi)=sumlimits_{i=1}^{r-1} x_ix_{i+1}, quad xiin mathbb{F}_q. $$ For a non-constant polynomial $f(X)in mathbb{F}_q[X]$ and $cin mathbb{F}_p$ we study the number of solutions $xiin mathbb{F}_q$ of $R(f(xi))=c$. If the degree $d$ of $f(X)$ is fixed, $rge 6$ and $prightarrow infty$, the number of solutions is asymptotically $p^{r-1}$ for any $c$. The proof is based on the Hooley-Katz Theorem." @default.
- W4287764327 created "2022-07-26" @default.
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- W4287764327 date "2020-06-04" @default.
- W4287764327 modified "2023-10-18" @default.
- W4287764327 title "On the distribution of the Rudin-Shapiro function for finite fields" @default.
- W4287764327 doi "https://doi.org/10.48550/arxiv.2006.02791" @default.
- W4287764327 hasPublicationYear "2020" @default.
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