Sumsets in vector spaces over finite fields

JOURNAL OF NUMBER THEORY

卷:71

Article

Eliahou, S ; Kervaire, M

关键词: additive number theory;    sumset;    restricted sumset;    polynomial method;    Cauchy-Davenport theorem;    Yuzvinsky theorem;    Erdos-Heilbronn conjecture;    Hopf-Stiefel-Pfister function;    Nim sum;    p-adic Nim sum;   

DOI  :  10.1006/jnth.1998.2235

来源: Elsevier

【 摘 要 】

We determine explicitly the least possible size of the sumset of two subsets A, B subset of (Z/pZ)(N) with fixed cardinalities, thereby generalizing both Cauchy-Davenport's theorem (case N = 1) and Yuzvinsky's theorem(case p = 2). The solution involves a natural generalization of the well-known Hopf-Stiefel-Pfister function. The corresponding problem for more than two summands is also considered and solved. We then consider restricted sumsets, formed by taking sums of distinct elements only. We determine almost completely the least possible size of the restricted sumset of two subsets in (Z/pZ)(N) with fixed cardinalities. Our result generalizes the recent solution(s) of the Erdos-Heilbronn conjecture dealing with the restricted sumsets of two equal subsets in Z/pZ. (C) 1998 Academic Press.

【 授权许可】

Free