A Pollard Type Result for Restricted Sums Original Research Article

Let image be an arbitrary field. Letpbe the characteristic of image in case of finite characteristic and ∞ if image has characteristic 0. LetAbe a finite subset of image. By logical and2 Awe denote the set {a+b a, bset membership, variantAanda≠b}. Forcset membership, variantlogical and2 A, letν(R)cbe one-half of the cardinality of the set of pairs (a, b) satisfyinga≠banda+b=c. Denote byμ(R)ithe cardinality of the set {cset membership, variantlogical and2 A ν(R)cgreater-or-equal, slantedi}. We prove that, fort=1, …, left floorA/2right floor, ∑ti=1 μ(R)igreater-or-equal, slantedt min{p, 2(A−t)−1}. For image=imagepandt=1 we get the Erdimages–Heilbronn conjecture, first proved by J. A. Dias da Silva and Y. O. Hamidoune (Bull. London Math. Soc.26, 1994, 140–146).