Explicit constructions of RIP matrices and related problems

We give a new explicit construction of $n\times N$ matrices satisfying the Restricted Isometry Property (RIP). Namely, for some c>0, large N and any n satisfying N^{1-c} < n < N, we construct RIP matrices of order k^{1/2+c}. This overcomes the natural barrier k=O(n^{1/2}) for proofs based on small coherence, which are used in all previous explicit constructions of RIP matrices. Key ingredients in our proof are new estimates for sumsets in product sets and for exponential sums with the products of sets possessing special additive structure. We also give a construction of sets of n complex numbers whose k-th moments are uniformly small for 1\le k\le N (Turan's power sum problem), which improves upon known explicit constructions when (\log N)^{1+o(1)} \le n\le (\log N)^{4+o(1)}. This latter construction produces elementary explicit examples of n by N matrices that satisfy RIP and whose columns constitute a new spherical code; for those problems the parameters closely match those of existing constructions in the range (\log N)^{1+o(1)} \le n\le (\log N)^{5/2+o(1)}.Comment: v3. Minor correction

Analytic Number Theory

Analytic number theory is a subject which is central to modern mathematics. There are many important unsolved problems which have stimulated a large amount of activity by many talented researchers. At least two of the Millennium Problems can be considered to be in this area. Moreover in recent years there has been very substantial progress on a number of these questions

On the Polynomial Szemer\'edi Theorem in Finite Commutative Rings

The polynomial Szemer\'{e}di theorem implies that, for any $\delta \in (0,1)$, any family $\{P_1,\ldots, P_m\} \subset \mathbb{Z}[y]$ of nonconstant polynomials with constant term zero, and any sufficiently large $N$, every subset of $\{1,\ldots, N\}$ of cardinality at least $\delta N$ contains a nontrivial configuration of the form $\{x,x+P_1(y),\ldots, x+P_m(y)\}$. When the polynomials are assumed independent, one can expect a sharper result to hold over finite fields, special cases of which were proven recently, culminating with arXiv:1802.02200, which deals with the general case of independent polynomials. One goal of this article is to explain these theorems as the result of joint ergodicity in the presence of asymptotic total ergodicity. Guided by this concept, we establish, over general finite commutative rings, a version of the polynomial Szemer\'{e}di theorem for independent polynomials $\{P_1,\ldots, P_m\} \subset \mathbb{Z}[y_1,\ldots, y_n]$, deriving new combinatorial consequences, such as the following. Let $\mathcal R$ be a collection of finite commutative rings subject to a mild condition on their torsion. There exists $\gamma \in (0,1)$ such that, for every $R \in \mathcal R$, every subset $A \subset R$ of cardinality at least $|R|^{1-\gamma}$ contains a nontrivial configuration $\{x,x+P_1(y),\ldots, x+P_m(y)\}$ for some $(x,y) \in R \times R^n$, and, moreover, for any subsets $A_0,\ldots, A_m \subset R$ such that $|A_0|\cdots |A_m| \geq |R|^{(m+1)(1-\gamma)}$, there is a nontrivial configuration $(x, x+P_1(y), \ldots, x+P_m(y)) \in A_0\times \cdots \times A_m$. The fact that general rings have zero divisors is the source of many obstacles, which we overcome; for example, by studying character sums, we develop a bound on the number of roots of an integer polynomial over a general finite commutative ring, a result which is of independent interest.Comment: 110 page