https://zbmath.org/atom/cc/11T30 2024-09-27T17:47:02.548271Z Werkzeug https://zbmath.org/1541.11068 2024-09-27T17:47:02.548271Z "Iyer, Siddharth" https://zbmath.org/authors/?q=ai:iyer.siddharth-s|iyer.siddharth "Shparlinski, Igor E." https://zbmath.org/authors/?q=ai:shparlinski.igor-e For a basis \(\{v_1, v_2, \ldots, v_r\}\) of \(\mathbb{F}_{q^r}\) over \(\mathbb{F}_q\) and \(\mathcal{A} \subseteq \mathbb{F}_q\), consider the set \(S_r(\mathcal{A}) := \{a_1v_1+a_2v_2+ \cdots +a_rv_r : a_1, a_2, \ldots, a_r \in \mathbb{F}_q\}\), i.e., that is the set of \(u \in \mathbb{F}_{q^r}\) whose coordinates are restricted to the set \(\mathcal{A}\). In particular, one of the natural examples is the case of \( q = 3\) and \(\mathcal{A} = \{0, 2\}\) which leads to a Cantor-like set \(S_r(\mathcal{A}) \subseteq \mathbb{F}_{q^r}\). In this article, the authors estimate the mixed character sums \[ S_r(\mathcal{A}; \chi, \psi; f_1, f_2) = \underset{\omega \in S_r(\mathcal{A})}{\sum}\chi(f_1(\omega))\psi(f_2(\omega)) \] with rational functions \(f_1(x), f_2(x) \in \mathbb{F}_{q^r}(x) \), of degrees \(d_1\) and \(d_2\), respectively and \(\chi\) and \(\psi\) are fixed multiplicative and additive characters of \(\mathbb{F}_{q^r}\), respectively (with the natural conventions that the poles of \(f_1(x)\) and \(f_2(x)\) are excluded from summation). In particular, they obtain a nontrivial estimate for such a sum over a finite field analogue of the Cantor set. Reviewer: Dhiren Kumar Basnet (Napaam) https://zbmath.org/1541.12002 2024-09-27T17:47:02.548271Z "Rani, Mamta" https://zbmath.org/authors/?q=ai:rani.mamta "Sharma, Avnish K." https://zbmath.org/authors/?q=ai:sharma.avnish-k "Tiwari, Sharwan K." https://zbmath.org/authors/?q=ai:tiwari.sharwan-kumar "Panigrahi, Anupama" https://zbmath.org/authors/?q=ai:panigrahi.anupama Let \(\mathbb{F}_q\) be a finite field of order \(q\) and \(\mathbb{F}_{q^n}\) be its extension of degree \(n\). In this article the existence of elements \(\alpha\) in finite fields \(\mathbb{F}_{q^n}\) such that both \(\alpha\) and its inverse \(\alpha^{-1}\) are \(r\)-primitive and \(k\)-normal over \(\mathbb{F}_q\) has been discussed. A characteristic function for the set of \(k\)-normal elements has been defined and it is used to establish a sufficient condition for the existence of the desired pair \((\alpha, \alpha^{-1})\). Also, it is shown that for \(n \geq 7\), there always exists a pair \((\alpha, \alpha^{-1})\) of 1-primitive and 1-normal elements in \(\mathbb{F}_{q^n}\) over \(\mathbb{F}_q\). Additionally, as an example, the authors have obtained that for \(n = 5, 6\), if \(\gcd(q, n) = 1\), there always exists such a pair in \(\mathbb{F}_{q^n}\), except for the field \(\mathbb{F}_{4^5}\). Reviewer: Dhiren Kumar Basnet (Napaam)