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.

Abstract. We study the k-resultant modulus set problem in the ddimensional vector space F d q over the finite field Fq with q elements.

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.

Abstract. In this paper we study the cardinality of the dot product set generated by two subsets of vector spaces over finite fields. We notice that the results on the dot product problems for one set can be simply extended to two sets. Let E and F be subsets of the d-dimensional vector space F d q over a finite field F q with q elements. As a new result, we prove that if E and F are subsets of the paraboloid and |E||F | ≥ Cq d for some large C > 1, then |Π(E, F )| ≥ cq for some 0 < c < 1. In particular, we find a connection between the size of the dot product set and the number of lines through both the origin and a nonzero point in the given set E. As an application of this observation, we obtain more sharpened results on the generalized dot product set problems. The discrete Fourier analysis and geometrical observation play a crucial role in proving our results.

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