Additive dimension and the growth of sets

This is a survey of old and new problems and results in additive number theory.

We study the relations between several notions of dimension for an additive set, some of which are well-known and some of which are more recent, appearing for instance in work of Schoen and Shkredov. We obtain bounds for the ratios between these dimensions by improving an inequality of Lev and Yuster, and we show that these bounds are asymptotically sharp, using in particular the existence of large dissociated subsets of $\{0,1\}^n\subset \mathbb{Z}^n$.

We prove some new bounds for the size of the maximal dissociated subset of structured (having small sumset, large energy and so on) subsets A of an abelian group.

We improve a result of Solymosi on sum-products in R, namely, we prove that max{|A+A|,|AA|}\gg |A|^{4/3+c}, where c>0 is an absolute constant. New lower bounds for sums of sets with small product set are found. Previous results are improved effectively for sets A from R with |AA| \le |A|^{4/3}.

We prove that finite sets of real numbers satisfying $|AA| \leq |A|^{1+\epsilon}$ with sufficiently small $\epsilon > 0$ cannot have small additive bases nor can they be written as a set of sums $B+C$ with $|B|, |C| \geq 2$. The result can be seen as a real analog of the conjecture of S\'ark\"ozy that multiplicative subgroups of finite fields of prime order are additively irreducible.

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.

This paper describes problems concerning the range of cardinalities of sumsets and restricted sumsets of finite subsets of the integers and finite subsets of ordered abelian groups.

We prove new quantitative bounds on the additive structure of sets obeying an $L^3$ 'control' assumption, which arises naturally in several questions within additive combinatorics. This has a number of applications - in particular we improve the known bounds for the sum-product problem, the Balog-Szemer\'{e}di-Gowers theorem, and the additive growth of convex sets.

We prove new results on additive properties of finite sets $A$ with small multiplicative doubling $|AA|\leq M|A|$ in the category of real/complex sets as well as multiplicative subgroups in the prime residue field. The improvements are based on new combinatorial lemmata, which may be of independent interest. Our main results are the inequality $$ |A-A|^3|AA|^5 \gtrsim |A|^{10}, $$ over the reals, "redistributing" the exponents in the textbook Elekes sum-product inequality a...

It was asked by E. Szemer\'edi if, for a finite set $A\subset\mathbb{Z}$, one can improve estimates for $\max\{|A+A|,|A\cdot A|\}$, under the constraint that all integers involved have a bounded number of prime factors -- that is, each $a\in A$ satisfies $\omega(a)\leq k$. In this paper, answer Szemer\'edi's question in the affirmative by showing that this maximum is of order $|A|^{\frac{5}{3}-o(1)}$ provided $k\leq (\log|A|)^{1-\epsilon}$ for some $\epsilon>0$. In fact, this...