| Let F be an arbitrary field. Let p be the characteristic of F in the case of finite characteristic and infinity if F has characteristic 0. Let A and B be nonempty finite subsets of F. For c ∈ F, let nuc(A, B) be the cardinality of the set of pairs (a, b) such that a + b = c, and mu i(A, B) the cardinality of the set of elements c ∈ A + B for which nu c(A, B) is greater than or equal to i.; In [6] Caldiera and Dias Da Silva proved the following theorem: Theorem 1. Let A and B be finite nonempty subsets of F. Then for t = 1,2,...,min{lcub}p, |A| + |B|{rcub} we have i=1tmi A,B≥t minp, A+B -t. This result is an extension to an arbitrary field of a theorem proved by Pollard, for F = Zp = Z/pZ, where p is a prime number. Notice that the case where t = 1 is well known as Cauchy-Davenport Theorem.; In [2] Caldiera and Dias Da Silva proved the following results, for restricted sums, as an analogue of Theorem 1.; Let A be a finite subset of F. We denote by Λ2 A the set a+b&vbm0;a, b∈Aanda≠b .; For c ∈ &Lgr;2A, let n2 c=12&vbm0; a,b∈A2,a ≠banda+b=c &vbm0;; and m2 i=&vbm0;&cubl0;c∈L2A &vbm0;n2 c≥i&cubr0;&vbm0;.; Then, for t=1,&ldots;,&fll0;&vbm0;A&vbm0;&solm0;2&flr0;, i=ttm 2i≥t minp,2 A-t-1 . This lower bound is tight and the equality is attained when A is an arithmetic progression.; For F = Zp and t = 1 we get the Erdos-Heilbronn conjecture.; In this paper I generalize this result to the restricted sum Λ hA for 2 ≤ h ≤ |A| ≤ p. |
