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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
Abstract

For A , let A + A = {a + b : a,b A} and AA = {ab : a,b A}. For k , let SP (k) denote the minimum value of max {|A + A|,|AA|} over all A with |A| = k. Here we establish SP (k) = 3k 3 for 2 k 7, the k = 7 case achieved for example by {1,2,3,4,6,8,12}, while SP (k) = 3k 2 for k = 8,9, the k = 9 case achieved for example by {1,2,3,4,6,8,9,12,16}. For 4 k 7, we provide two proofs using different applications of Freiman’s 3k4 theorem; one of the proofs includes extensive case analysis on the product sets of k-element subsets of (2k3)-term arithmetic progressions. For k = 8,9, we apply Freiman’s 3k3 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 r2 and r 2.

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