On Combinatorial Complexity of Convex Sequences

We show that the equation si1 + si2 + · · ·+ sid = sid+1 + · · ·+ si2d has O N2d−2+2 −d+1 solutions for any strictly convex sequence {si}i=1 without any additional arithmetic assumptions. The proof is based on weighted incidence theory and an inductive procedure which allows us to effectively deal with higher dimensional interactions. The terminology ”combinatorial complexity” is borrowed from [CEGSW90] where much of our higher dimensional incidence theoretic motivation comes from. Section 1: Introduction and statement of results Consider a sequence of real numbers {si}Ni=1. It is a classical problem in number theory to determine the number Nd = Nd(N) of solutions of the equation (1.1) si1 + si2 + · · ·+ sid = sid+1 + · · ·+ si2d . The number of solutions Nd will certainly depend on geometric and arithmetic properties of the sequence {si}. A trivial example is if si = i, when the number of solutions of (1.1) is approximately N2d−1. Here and throughout the paper the notations a . b, or a = O(b) means that there exists C > 0 such that a ≤ Cb, and a ≈ b means that a . b and b . a. Besides, a / b, with respect to a large parameter N , means that for every 2 > 0 there exists C2 > 0 such that a ≤ C2N b. 1991 Mathematics Subject Classification. Primary 11D45, 11L07; Secondary 52B55.