Discrete & Computational Geometry

, Volume 35, Issue 1, pp 143–158 | Cite as

Combinatorial Complexity of Convex Sequences

  • Alex Iosevich
  • S. Konyagin
  • M. Rudnev
  • V. Ten


We show that the equation \[ s_{i_1}+s_{i_2}+\cdots+s_{i_d}=s_{i_{d+1}}+\cdots+s_{i_{2d}} \] has $O(N^{2d-2+2^{-d+1}})$ solutions for any strictly convex sequence $\{s_i\}_{i=1}^N$ without any additional arithmetic assumptions. The proof is based on weighted incidence theory and an inductive procedure which allows us to deal with higher-dimensional interactions effectively. The terminology "combinatorial complexity" is borrowed from [CES+] where much of our higher-dimensional incidence theoretic motivation comes from.


Computational Mathematic Combinatorial Complexity Inductive Procedure Theoretic Motivation Additional Arithmetic 
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Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Department of Mathematics, University of Missouri, Columbia, MO 65201USA

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