The sum-product problem for small sets

TL;DR

The paper determines the exact values of $SP(k)$ for small $k$ in the natural numbers, proving $SP(k)=3k-3$ for $2\le k\le 7$ and $SP(k)=3k-2$ for $k=8,9$, with explicit examples achieving these bounds. It develops two independent Freiman-based techniques to prove $SP(k)\ge 3k-3$ for $4\le k\le 7$, one via the sum-side structure forcing an arithmetic progression and one via a product-side argument forcing a geometric progression, which together imply the same lower bound. For $k\ge 8$, Freiman's $3k-3$ and $3k-3$-type classifications yield that near-extremal sets are unions of two geometric progressions with the same ratio, and a detailed overlap analysis (split into $r\neq 2$ and $r\ge 2$) shows $|A+A|\ge 3k-2$, establishing the bound. Concrete $k=2$ to $k=9$ examples are provided, and the paper discusses extending the exact small-$k$ results (notably the $SP(10)$ case) and directions for future work.

Abstract

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

Figures (4)

• Figure 1: The red path corresponds to a strictly increasing sequence of $15$ elements of $AA$ when $k=6$ and $\tilde{A}=\{0,4,5,6,7,8\}$. This is one of ${8 \choose 5}=56$ cases for $k=6$. In this and most other cases, longer qualifying paths exist, but we do not prioritize optimality in our analysis, only reaching $3k-3$.
• Figure 2: An illustration of $(B+B)\cap(C+C)=\{4,12,36\}$, where $B=\{1,3,9,27\}$ and $C=\{2,6,18,54\}$. Each color shows a member of the single geometric family of solutions, stemming from the solution $3+1=2+2$.
• Figure 3: An illustration of $(B+B)\cap(B+C)=\{24,36,54\}$, where $B=\{8,12,18,27\}$ and $C=\{16,24,36,54\}$. Each color shows a member of the single geometric family of solutions, stemming from the solution $12+12=8+16$.
• Figure 4: An illustration showing that if $B=\{1,3,9,27\}$ and $C=\{4,12,36,108\}$, then the only elements of $B+C$ with two distinct representations are $13$ and $39$, stemming from the solution $9-1=12-4$. In particular, $|B+C|=(4)(4)-2=14$.

Theorems & Definitions (22)

• Theorem 1.1
• Theorem 2.1: Freiman's $3k-4$ theorem
• Corollary 2.2
• proof : Proof
• Lemma 2.3: Rational root theorem
• Lemma 2.4
• proof
• Corollary 2.5
• Theorem 3.1: Freiman's $3k-3$ theorem
• Corollary 3.2