On the Sums of Any $k$ Points in Finite Fields

Abstract

Abstract. For a set E ⊂ F d q , we define the k-resultant magnitude set asIn this paper we find a connection between a lower bound of the cardinality of the k-resultant magnitude set and the restriction theorem for spheres in finite fields. As a consequence, it is shown that if +ε for ε > 0, then |∆ 3 (E)| ≥ cq.

The Generalized k-Resultant Modulus Set Problem in Finite Fields

“…. , E k ), is defined byfor d = 4, 6 with a sufficiently large constantThis generalizes the previous result in [5]. We also show that ifThis result improves the previous work in [5] by removing ε > 0 from the exponent.…”

“…However, in higher even dimensions d ≥ 4, the exponent (d + 1)/2 has not been improved. To demonstrate some possibility that the exponent (d + 1)/2 could be improved for even dimensions d ≥ 4, the authors in [5] introduced a k-resultant modulus set which generalizes the distance set in the sense that any k points can be selected from a set E ⊂ F d q to determine an object similar to a distance. More precisely, for a set E ⊂ F d q we define a k-resultant modulus set ∆ k (E) as Question 1.2.…”

The k-resultant modulus set problem on algebraic varieties over finite fields

“…Likewise, it was shown in [3] that β 2 (d) = d+1 2 . In view of these examples, the following conjecture was given by the authors in [4].…”

Distinct distances on regular varieties over finite fields

“…Let D(x) = x There are various papers studying the cardinality of ∆(E), see for example [3,9,5,4,10] and references therein. In this paper, we are interested in the case when E is a subset in a regular variety.…”