The Generalized k-Resultant Modulus Set Problem in Finite Fields

Abstract

Let F d q be the d-dimensional vector space over the finite field Fq with q elements. Given k sets E j ⊂ F d q for j = 1, 2, . . . , k, the generalized k-resultant modulus set, denoted by ∆ k (E 1 , E 2 , . . . , 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.

A spherical extension theorem and applications in positive characteristic

“…This definition should be compared with the definition of the Fourier transform f used in other papers. We emphasize that there does not appear a normalizing factor q −d in the definition of f , while such a normalizing factor has been used in many other articles such as [12], [7], [1], and [2]. Recall that the Fourier inversion theorem states that…”

“…For β ∈ F q , let P β be a translate of P by β defined by x d + β = x 2 1 + • • • + x 2 d−1 . For j = 0, let S j be the sphere centered at the origin of radius j, namely, S j = {(x 1 , .…”

Extension theorems in vector spaces over finite fields

“…Compute the first term above by applying the first equation in (13) and using the fact that r =0 χ(jr) = −1 for j = 0. Then we see that…”

“…In order to compute the first term above, we use the basic Gauss sum estimates in (13). To compute the second term above, we notice that the sum over r = 0 is −1, because j = 0.…”