[Paper Review] The Erdős-Szemerédi problem on sum set and product set

Mei-Chu Chang|arXiv (Cornell University)|Feb 17, 2004

Limits and Structures in Graph Theory5 references4 citations

TL;DR

This paper resolves a key conjecture in additive combinatorics by establishing the asymptotic growth rate of the minimal sum of the sizes of the sumset and product set over all $k$-element subsets of integers. Using a novel approach combining Freiman's theorem, trigonometric polynomial estimates, and Ruzsa's Pl"unnecke-type inequalities, the authors prove that $\ln g(k) \sim \frac{(\ln k)^2}{\ln \ln k}$, confirming the Erd\'os-Szemer\'edi conjecture on the superpolynomial growth of $g(k)$.

ABSTRACT

The basic theme of this paper is the fact that if $A$ is a finite set of integers, then the sum and product sets cannot both be small. A precise formulation of this fact is Conjecture 1 below due to Erd\H os-Szemerédi [E-S]. (see also [El], [T], and [K-T] for related aspects.) Only much weaker results or very special cases of this conjecture are presently known. One approach consists of assuming the sum set $A + A$ small and then deriving that the product set $AA$ is large (using Freiman's structure theorem). (cf [N-T], [Na3].) We follow the reverse route and prove that if $|AA| < c|A|$, then $|A+A| > c^\prime |A|^2$ (see Theorem 1). A quantitative version of this phenomenon combined with Plünnecke type of inequality (due to Ruzsa) permit us to settle completely a related conjecture in [E-S] on the growth in $k$. If $$ g(k) \equiv ext{min}\{|A[1]| + |A\{1\}|\} $$ over all sets $A\subset \Bbb Z$ of cardinality $|A| = k$ and where $A[1]$ (respectively, $A\{1\}$) refers to the simple sum (resp., product) of elements of $A$. (See (0.6), (0.7).) It was conjectured in [E-S] that $g(k)$ grows faster than any power of $k$ for $k o\infty$. We will prove here that $\ell n g(k)\sim\frac{(\ell n k)^2}{\ell n \ell n k}$ (see Theorem 2) which is the main result of this paper.

Motivation & Objective

• To resolve Conjecture 2 of Erd\'os and Szemer\'edi on the superpolynomial growth of $g(k) = \min \{ |A[1]| + |A\{1\}| \} $ over all $k$-element sets $A \subset \mathbb{N}$.
• To establish a quantitative lower bound on the size of the sumset $2A$ when the product set $A^2$ is small, via a new method involving trigonometric polynomial estimates.
• To apply Pl\
• To derive a precise asymptotic formula for $\ln g(k)$, confirming the conjectured growth rate of $g(k)$ as $k \to \infty$.
• To introduce and analyze the concept of multiplicative dimension of finite sets to control the structure of product sets and support the main estimates.

Proposed method

• Applying a refined version of Freiman's theorem to show that if $|A^2| < \alpha |A|$, then $A$ is contained in a generalized geometric progression of controlled size and multiplicative dimension.
• Using trigonometric polynomial estimates in the spirit of Rudin to bound the $L^{2h}$-norm of the exponential sum $\left| \sum_{a \in A} e^{2\pi i a x} \right|^{2h}$, which controls the size of $hA$.
• Introducing the concept of multiplicative dimension of a finite set to quantify the structural complexity of product sets and derive uniform bounds on trigonometric sums.
• Combining the sumset lower bound from Theorem 1 with Ruzsa's Pl\
• Using the result of Laczkovich and Ruzsa on the number of homothetic subsets to bound the size of $A[1]$ and $A\{1\}$ in terms of the multiplicative dimension and logarithmic factors.
• Applying double counting and logarithmic estimates to derive the final asymptotic for $g(k)$, balancing the growth of sumset and product set sizes.

Experimental results

Research questions

• RQ1 What is the precise asymptotic growth rate of $g(k) = \min \{ |A[1]| + |A\{1\}| \} $ over all $k$-element subsets $A \subset \mathbb{N}$?
• RQ2 Can the conjecture that $g(k)$ grows faster than any polynomial in $k$ be confirmed, and if so, what is the exact rate?
• RQ3 How does the size of the sumset $2A$ behave when the product set $A^2$ is small, and can this be quantified?
• RQ4 What structural properties of finite sets of integers govern the interplay between sumset and product set growth?
• RQ5 Can the multiplicative dimension of a set be used as a tool to derive uniform bounds on exponential sums and hence on sumset sizes?

Key findings

• The paper proves that $\ln g(k) \sim \frac{(\ln k)^2}{\ln \ln k}$, which is the main result and confirms the Erd\'os-Szemer\'edi conjecture on the superpolynomial growth of $g(k)$.
• It establishes a quantitative lower bound: if $|A^2| < \alpha |A|$, then $|2A| > 36^{-\alpha} |A|^2$, showing that a small product set forces a large sumset.
• The authors derive a general lower bound $|hA| > c_h(\alpha) |A|^h$ with $c_h(\alpha) = (2h^2 - h)^{-h\alpha}$, which holds when $|A^2| < \alpha |A|$.
• The upper bound for $g(k)$ is shown to be $g(k) < k^{(1+\varepsilon) \frac{\ln k}{\ln \ln k}}$ for any $\varepsilon > 0$ and sufficiently large $k$, matching the lower bound up to the constant in the exponent.
• The lower bound for $g(k)$ is improved to $g(k) > k^{(\frac{1}{2} - \varepsilon) \frac{\ln k}{\ln \ln k}}$ under the assumption of Ruzsa's refinement, which is stronger than the initial $\frac{1}{8}$-bound.
• The paper introduces and develops the concept of multiplicative dimension of a finite set, which is used to control the structure of product sets and to derive uniform estimates on exponential sums.