SIZE OF DOT PRODUCT SETS DETERMINED BY PAIRS OF SUBSETS OF VECTOR SPACES OVER FINITE FIELDS

• 발행기관: 충청수학회
• 저자명: Doowon Koh Youngjin Pi
• 간행물 정보: 『Journal of the Chungcheong Mathematical Society』Volume 26, No. 4, 853~867쪽, 전체 15쪽
• 주제분류: 자연과학 > 자연과학일반
• 파일형태: PDF
• 발행일자: 2013.11.30

영문 초록

In this paper we study the cardinality of the dot prod-uct set generated by two subsets of vector spaces over ¯nite ¯elds. 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 Fd q over a ¯nite ¯eld Fq with q elements. As a new result, we prove that if E and F are subsets of the parab-oloid and jEjjFj ¸ Cqd for some large C > 1; then j¦(E; F)j ¸ cq for some 0 < c < 1: In particular, we ¯nd 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 gener- alized dot product set problems. The discrete Fourier analysis and geometrical observation play a crucial role in proving our results.

목차

1. Introduction

2. Preliminaries

3. Results on the generalized dot product sets

4. Sharpened results on the generalized dot product set