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arXiv · 2304.00888

Local Differences Determined by Convex sets

Krishnendu Bhowmick·Miriam Patry·Oliver Roche-Newton
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Abstract

This paper introduces a new problem concerning additive properties of convex sets. Let $S= \{s_1 < \dots <s_n \}$ be a set of real numbers and let $D_i(S)= \{s_x-s_y: 1 \leq x-y \leq i\}$. We expect that $D_i(S)$ 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 $D_3(S)$ 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 $D_4(S)$, namely $|D_4(S)| \geq \frac{5}{4}n -1$. For sufficiently large values of $i$, we are able to prove a non-trivial bound that grows with $i$ using incidence geometry.

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Krishnendu Bhowmick, Miriam Patry, Oliver Roche-Newton. 2023-04-03. Local Differences Determined by Convex sets. https://arxiv.org/abs/2304.00888

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