New solutions to the Hurwitz problem on square identities

The Hurwitz problem of composition of quadratic forms, or of "sum of squares identity" is tackled with the help of a particular class of $(\mathbb{Z}_2)^n$-graded non-associative algebras generalizing the octonions. This method provides an explicit formula for the classical Hurwitz-Radon identity and leads to new solutions in a neighborhood of the Hurwitz-Radon identity.Comment: 13 pages, 2 figures, final version to appear in J. Pure Appl. Al

On the Bogolyubov-Ruzsa lemma

Our main result is that if A is a finite subset of an abelian group with |A+A| < K|A|, then 2A-2A contains an O(log^{O(1)} K)-dimensional coset progression M of size at least exp(-O(log^{O(1)} K))|A|.Comment: 28 pp. Corrected typos. Added appendix on model settin

A monad measure space for logarithmic density

We provide a framework for proofs of structural theorems about sets with positive Banach logarithmic density. For example, we prove that if $A\subseteq \mathbb{N}$ has positive Banach logarithmic density, then $A$ contains an approximate geometric progression of any length. We also prove that if $A,B\subseteq \mathbb{N}$ have positive Banach logarithmic density, then there are arbitrarily long intervals whose gaps on $A\cdot B$ are multiplicatively bounded, a multiplicative version Jin's sumset theorem. The main technical tool is the use of a quotient of a Loeb measure space with respect to a multiplicative cut.Comment: 26 page

Kakeya sets of curves

We investigate analogues for curves of the Kakeya problem for straight lines. These arise from H"ormander-type oscillatory integrals in the same way as the straight line case comes from the restriction and Bochner-Riesz problems. Using some of the geometric and arithmetic techniques developed for the straight line case by Bourgain, Wolff, Katz and Tao, we are able to prove positive results for families of parabolas whose coefficients satisfy certain algebraic conditions.Comment: 32 pages. To appear in GAF

Top-k-Convolution and the Quest for Near-Linear Output-Sensitive Subset Sum

In the classical Subset Sum problem we are given a set $X$ and a target $t$, and the task is to decide whether there exists a subset of $X$ which sums to $t$. A recent line of research has resulted in $\tilde{O}(t)$-time algorithms, which are (near-)optimal under popular complexity-theoretic assumptions. On the other hand, the standard dynamic programming algorithm runs in time $O(n \cdot |\mathcal{S}(X,t)|)$, where $\mathcal{S}(X,t)$ is the set of all subset sums of $X$ that are smaller than $t$. Furthermore, all known pseudopolynomial algorithms actually solve a stronger task, since they actually compute the whole set $\mathcal{S}(X,t)$. As the aforementioned two running times are incomparable, in this paper we ask whether one can achieve the best of both worlds: running time $\tilde{O}(|\mathcal{S}(X,t)|)$. In particular, we ask whether $\mathcal{S}(X,t)$ can be computed in near-linear time in the output-size. Using a diverse toolkit containing techniques such as color coding, sparse recovery, and sumset estimates, we make considerable progress towards this question and design an algorithm running in time $\tilde{O}(|\mathcal{S}(X,t)|^{4/3})$. Central to our approach is the study of top-$k$-convolution, a natural problem of independent interest: given sparse polynomials with non-negative coefficients, compute the lowest $k$ non-zero monomials of their product. We design an algorithm running in time $\tilde{O}(k^{4/3})$, by a combination of sparse convolution and sumset estimates considered in Additive Combinatorics. Moreover, we provide evidence that going beyond some of the barriers we have faced requires either an algorithmic breakthrough or possibly new techniques from Additive Combinatorics on how to pass from information on restricted sumsets to information on unrestricted sumsets

Entropy and set cardinality inequalities for partition-determined functions

A new notion of partition-determined functions is introduced, and several basic inequalities are developed for the entropy of such functions of independent random variables, as well as for cardinalities of compound sets obtained using these functions. Here a compound set means a set obtained by varying each argument of a function of several variables over a set associated with that argument, where all the sets are subsets of an appropriate algebraic structure so that the function is well defined. On the one hand, the entropy inequalities developed for partition-determined functions imply entropic analogues of general inequalities of Pl\"unnecke-Ruzsa type. On the other hand, the cardinality inequalities developed for compound sets imply several inequalities for sumsets, including for instance a generalization of inequalities proved by Gyarmati, Matolcsi and Ruzsa (2010). We also provide partial progress towards a conjecture of Ruzsa (2007) for sumsets in nonabelian groups. All proofs are elementary and rely on properly developing certain information-theoretic inequalities.Comment: 26 pages. v2: Revised version incorporating referee feedback plus inclusion of some additional corollaries and discussion. v3: Final version with minor corrections. To appear in Random Structures and Algorithm