The sum-product problem for small sets

Clevenger, Ginny Ray; Havard, Haley; Heard, Patch; Lott, Andrew; Rice, Alex; Wilson, Brittany

doi:10.2140/involve.2025.18.165

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The sum-product problem for small sets

Ginny Ray Clevenger, Haley Havard, Patch Heard, Andrew Lott, Alex Rice and Brittany Wilson

Vol. 18 (2025), No. 1, 165–180

DOI: 10.2140/involve.2025.18.165

Abstract

For $A \subseteq ℝ$ , let $A + A = {a + b : a, b \in A}$ and $A A = {a b : a, b \in A}$ . For $k \in ℕ$ , let $SP (k)$ denote the minimum value of $\max {| A + A |, | A A |}$ over all $A \subseteq ℕ$ with $| A | = k$ . Here we establish $SP (k) = 3 k - 3$ for $2 \leq k \leq 7$ , the $k = 7$ case achieved for example by ${1, 2, 3, 4, 6, 8, 12}$ , while $SP (k) = 3 k - 2$ for $k = 8, 9$ , the $k = 9$ case achieved for example by ${1, 2, 3, 4, 6, 8, 9, 12, 16}$ . For $4 \leq k \leq 7$ , we provide two proofs using different applications of Freiman’s $3 k - 4$ theorem; one of the proofs includes extensive case analysis on the product sets of $k$ -element subsets of $(2 k - 3)$ -term arithmetic progressions. For $k = 8, 9$ , we apply Freiman’s $3 k - 3$ theorem for product sets, and investigate the sumset of the union of two geometric progressions with the same common ratio $r > 1$ , with separate treatments of the overlapping cases $r \neq 2$ and $r \geq 2$ .

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Keywords

sum-product problem, arithmetic combinatorics

Mathematical Subject Classification

Primary: 11B30

Milestones

Received: 28 July 2023

Revised: 12 August 2023

Accepted: 18 August 2023

Published: 24 January 2025

Communicated by Nathan Kaplan

Authors

Ginny Ray Clevenger
Millsaps College Jackson, MS United States

Haley Havard
Millsaps College Jackson, MS United States

Patch Heard
Millsaps College Jackson, MS United States

Andrew Lott
University of Georgia Athens, GA United States

Alex Rice
Department of Mathematics Millsaps College Jackson, MS United States

Brittany Wilson
Millsaps College Jackson, MS United States

Vol. 18, No. 1, 2025