Set systems with distinct sumsets

A family $\mathcal{A}$ of $k$-subsets of $\{1,2,\dots, N\}$ is a Sidon system if the sumsets $A+A'$, $A,A'\in \mathcal{A}$ are pairwise distinct. We show that the largest cardinality $F_k(N)$ of a Sidon system of $k$-subsets of $[N]$ satisfies $F_k(N)\le {N-1\choose k-1}+N-k$ and the asymptotic lower bound $F_k(N)=\Omega_k(N^{k-1})$. More precise bounds on $F_k(N)$ are obtained for $k\le 3$. We also obtain the threshold probability for a random system to be Sidon for $k= 2$ and $3$.Peer ReviewedPostprint (author's final draft

Sidon set systems

A family ${\mathcal A}$ of $k$-subsets of $\{1,2,\dots, N\}$ is a Sidon system if the sumsets $A+B$, $A,B\in \mathcal{A}$ are pairwise distinct. We show that the largest cardinality $F_k(N)$ of a Sidon system of $k$-subsets of $[N]$ satisfies $F_k(N)\le {N-1\choose k-1}+N-k$ and the asymptotic lower bound $F_k(N)=\Omega_k(N^{k-1})$. More precise bounds on $F_k(N)$ are obtained for $k\le 3$. We also obtain the threshold probability for a random system to be Sidon for $k\ge 2$.Comment: Incorporated referee comments. Published in Rev. Mat. Iberoa

On various restricted sumsets

For finite subsets A_1,...,A_n of a field, their sumset is given by {a_1+...+a_n: a_1 in A_1,...,a_n in A_n}. In this paper we study various restricted sumsets of A_1,...,A_n with restrictions of the following forms: a_i-a_j not in S_{ij}, or alpha_ia_i not=alpha_ja_j, or a_i+b_i not=a_j+b_j (mod m_{ij}). Furthermore, we gain an insight into relations among recent results on this area obtained in quite different ways.Comment: 11 pages; final version for J. Number Theor

A probabilistic technique for finding almost-periods of convolutions

We introduce a new probabilistic technique for finding 'almost-periods' of convolutions of subsets of groups. This gives results similar to the Bogolyubov-type estimates established by Fourier analysis on abelian groups but without the need for a nice Fourier transform to exist. We also present applications, some of which are new even in the abelian setting. These include a probabilistic proof of Roth's theorem on three-term arithmetic progressions and a proof of a variant of the Bourgain-Green theorem on the existence of long arithmetic progressions in sumsets A+B that works with sparser subsets of {1, ..., N} than previously possible. In the non-abelian setting we exhibit analogues of the Bogolyubov-Freiman-Halberstam-Ruzsa-type results of additive combinatorics, showing that product sets A B C and A^2 A^{-2} are rather structured, in the sense that they contain very large iterated product sets. This is particularly so when the sets in question satisfy small-doubling conditions or high multiplicative energy conditions. We also present results on structures in product sets A B. Our results are 'local' in nature, meaning that it is not necessary for the sets under consideration to be dense in the ambient group. In particular, our results apply to finite subsets of infinite groups provided they 'interact nicely' with some other set.Comment: 29 pages, to appear in GAF