On the Number of Solutions of a Linear Equation over Finite Sets

The largest possible number of representations of an integer in thek-fold sumsetkA=A+…+Ais maximal forAbeing an arithmetic progression. More generally, consider the number of solutions of the linear equationc1a1+…+ckak=λ, whereci≠0 andλare fixed integer coefficients, and where the variablesairange over finite sets of integersA1, …, Ak. We prove that for fixed cardinalitiesni=|Ai|, this number of solutions is maximal whenc1=…=ck=1,λ=0 and theAiare arithmetic progressions balanced around 0 and with the same common difference. For the corresponding residues problem, assumingci,λ∈FpandAi⊆Fp(where Fpis the set of residues modulo primep), the number of solutions of the equation above does not exceed1pn1…nk+8π n1…nkn21plus;…+n2k (1+o(1))ask→∞ and under some mild restrictions onni. This is best possible save for the constant in the second term: we conjecture that in fact 8 can be replaced by 6.