[Paper Review] A superadditivity and submultiplicativity property for cardinalities of sumsets

Katalin Gyarmati, Imre Z. Ruzsa+1 more|ArXiv.org|Jul 18, 2007

Limits and Structures in Graph Theory10 references4 citations

TL;DR

This paper establishes superadditivity and submultiplicativity inequalities for the cardinalities of sumsets in finite sets of integers and commutative semigroups. It proves that the size of an $n$-fold sumset $S = A_1 + \cdots + A_n$ satisfies $|S| \geq \frac{1}{k-1}\sum |S_i| - \frac{1}{k-1}$, where $S_i$ are $(n-1)$-fold sumsets omitting one set, and $|S| \leq \left(\prod |S_i|\right)^{1/(k-1)}$, showing subexponential growth. These results generalize Cauchy-Davenport and Lev-type inequalities to heterogeneous summands and extend to torsion-free groups and restricted addition graphs.

ABSTRACT

For finite sets of integers $A_1, A_2 ... A_n$ we study the cardinality of the $n$-fold sumset $A_1+... +A_n$ compared to those of $n-1$-fold sumsets $A_1+... +A_{i-1}+A_{i+1}+... A_n$. We prove a superadditivity and a submultiplicativity property for these quantities. We also examine the case when the addition of elements is restricted to an addition graph between the sets.

Motivation & Objective

• To establish a superadditivity property for the cardinality of $n$-fold sumsets when summands are finite sets of integers.
• To prove a submultiplicativity inequality for the growth of sumset sizes in commutative semigroups.
• To extend these results to torsion-free groups and to settings with restricted addition via addition graphs.
• To generalize known inequalities such as Cauchy-Davenport and Lev’s result on $|3A| \geq \frac{3}{2}|2A| - \frac{1}{2}$ to arbitrary finite sets with distinct summands.
• To investigate the behavior of $|kA|^{1/k}$ and confirm its monotonic decrease, supporting a conjecture by Ruzsa.

Proposed method

• Define $S = A_1 + \cdots + A_k$ and $S_i = A_1 + \cdots + A_{i-1} + A_{i+1} + \cdots + A_k$, the sumset omitting $A_i$.
• Construct $S_i'$ by replacing $A_i$ with its two-element subset $A_i'$ containing the minimal and maximal elements of $A_i$.
• Define $S' = \bigcup_{i=1}^k S_i'$, and prove $|S| \geq |S'| \geq \frac{1}{k-1}\sum |S_i| - \frac{1}{k-1}$ using combinatorial and extremal set arguments.
• Use Plünnecke-type inequalities and iterative applications of Theorem 4.2 to bound the size of sumsets with large subsets, ensuring the existence of a subset $X \subset A$ with controlled doubling.
• Apply Theorem 4.4 to find a subset $X$ of size greater than $t$ such that $|X + B|$ is bounded in terms of $s = \prod |A + B_i|$, with $B = B_1 + \cdots + B_h$.
• Use a scaling argument via $k$-fold direct products to eliminate a factor of 2 in intermediate bounds, ultimately proving $|S + A| \leq \sqrt{s|S|}$.

Experimental results

Research questions

• RQ1 Can the inequality $|3A| \geq \frac{3}{2}|2A| - \frac{1}{2}$ be extended to heterogeneous sumsets $A + B + C$?
• RQ2 Does the superadditivity inequality $|S| \geq \frac{1}{k-1}\sum |S_i| - \frac{1}{k-1}$ hold for arbitrary finite sets of integers?
• RQ3 Is the submultiplicativity bound $|S| \leq \left(\prod |S_i|\right)^{1/(k-1)}$ valid for sumsets in commutative semigroups?
• RQ4 Can the superadditivity and submultiplicativity properties be extended to torsion-free groups, where min/max elements are not defined?
• RQ5 Does the inequality $|S + A| \leq \sqrt{|A+B_1||A+B_2||S|}$ hold under restricted addition, such as on a graph?

Key findings

• The superadditivity inequality $|S| \geq \frac{1}{k-1}\sum_{i=1}^k |S_i| - \frac{1}{k-1}$ holds for finite sets of integers, with equality possible when sets are arithmetic progressions.
• The submultiplicativity bound $|S| \leq \left(\prod_{i=1}^k |S_i|\right)^{1/(k-1)}$ is proven for finite sets in commutative semigroups, generalizing known results for $k=3$.
• For torsion-free groups, a superadditivity result holds with $A_i'$ being two-element subsets containing minimal and maximal elements in a suitable ordering.
• The growth of $|kA|^{1/k}$ is shown to be non-increasing, confirming a conjecture by Ruzsa, using Plünnecke-type inequalities.
• The method of using $k$-fold direct products to eliminate a factor of 2 in intermediate bounds leads to the sharp inequality $|S + A| \leq \sqrt{s|S|}$.
• The results extend to addition graphs, though the submultiplicativity inequality fails in general for such restricted settings.