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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
Article

Abstract

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.

Keywords

Computational Mathematic Combinatorial Complexity Inductive Procedure Theoretic Motivation Additional Arithmetic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Copyright information

© Springer 2005

Authors and Affiliations

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

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