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- Title
LOCAL DIFFERENCES DETERMINED BY CONVEX SETS.
- Authors
Bhowmick, Krishnendu; Patry, Miriam; Roche-Newton, Oliver
- Abstract
This paper introduces a new problem concerning additive properties of convex sets. Let S = {s1 < ... < sn} be a set of real numbers and let Di(S) = {sx - sy: 1 ≤ x - y ≤ i}. We expect that Di(S) is large, with respect to the size of S and the parameter i, for any convex set S. We give a construction to show that D3(S) can be as small as n + 2, and show that this is the smallest possible size. On the other hand, we use an elementary argument to prove a non-trivial lower bound for D4(S), namely |D4(S)| ≥ 5/4n 1. For sufficiently large values of i, we are able to prove a non-trivial bound that grows with i using incidence geometry.
- Subjects
CONVEX sets; REAL numbers; RAMSEY numbers
- Publication
Integers: Electronic Journal of Combinatorial Number Theory, 2023, Vol 23, p1
- ISSN
1553-1732
- Publication type
Article
- DOI
10.5281/zenodo.8349093