Two Addition Theorems on Groups of Prime Order

✍ Scribed by W.D. Gao

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 162 KB
• Volume: 56
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1996.0013

✦ Synopsis

Let S=(a 1 , a 2 , ..., a 2n&1 ) be a sequence of 2n&1 elements in an Abelian group G of order n (written additively). For a # G, let r(S, a) be the number of subsequences of length exactly n whose sum is a. Erdo s et al. [1] proved that r(S, 0) 1. In [2], Mann proved that if n (=p) is a prime, then r(S, a) 1 for all a # G, provided no element is repeated more than p times in the sequence S. In this paper we prove the following generalization of Mann's theorem.

Proof of Theorem 1. (i) By Theorem 2 and Mann's theorem we have r(S, a) p for all 0{a # G.

(ii) If r(S, 0)<p+1, then by Theorem 2 we have r(S, 0)=1

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