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- W1951528160 abstract "We study the $k$-resultant modulus set problem in the $d$-dimensional vector space $mathbb F_q^d$ over the finite field $mathbb F_q$ with $q$ elements. Given $Esubset mathbb F_q^d$ and an integer $kge 2$, the $k$-resultant modulus set, denoted by $Delta_k(E)$, is defined as $$ Delta_k(E)={|x^1pm x^2 pm cdots pm x^k|in mathbb F_q: x^jin E, ~j=1,2,ldots, k},$$ where $|alpha|=alpha_1^2+cdots+ alpha_d^2$ for $alpha=(alpha_1, ldots, alpha_d) in mathbb F_q^d.$ In this setting, the $k$-resultant modulus set problem is to determine the minimal cardinality of $Esubset mathbb F_q^d$ such that $Delta_k(E) = mathbb F_q$ or $mathbb{F}_q^*$. This problem is an extension of the ErdH{o}s-Falconer distance problem. In particular, we investigate the $k$-resultant modulus set problem with the restriction that the set $Esubset mathbb F_q^d$ is contained in a specific algebraic variety. Energy estimates play a crucial role in our proof." @default.
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- W1951528160 date "2015-08-11" @default.
- W1951528160 modified "2023-09-24" @default.
- W1951528160 title "The $k$-resultant modulus set problem on algebraic varieties over finite fields" @default.
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- W1951528160 doi "https://doi.org/10.48550/arxiv.1508.02688" @default.
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