Sergei Konyagin

Explicit constructions of RIP matrices and related problems

We give a new explicit construction ofn×N matrices satisfying the Restricted Isometry Property (RIP). Namely, for some e > 0, largeN , and anyn satisfyingN1−e ≤ n ≤ N , we construct RIP matrices of order k ≥ n1/2+e and constant δ = n−e. This overcomes the natural barrier k = O(n1/2) 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 kth moments are uniformly small for 1 ≤ k ≤ N (Turan’s power sum problem), which improves upon known explicit constructions when (logN)1+o(1) ≤ n ≤ (logN)4+o(1). This latter construction produces elementary explicit examples of n×N matrices that satisfy the RIP and whose columns constitute a new spherical code; for those problems the parameters closely match those of existing constructions in the range (logN )1+o(1) ≤ n ≤ (logN)5/2+o(1).

On Sum Sets of Sets Having Small Product Set

We improve the sum–product result of Solymosi in R; namely, we prove that max{|A + A|, |AA|} ****** |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 ⊂ R with |AA| ≤ |A|4/3.

On congruences with products of variables from short intervals and applications

We obtain upper bounds on the number of solutions to congruences of the type (x1 + s)... (xν + s) ≡ (y1 + s)... (xν + s) ≢ 0 (mod p) modulo a prime p with variables from some short intervals. We give some applications of our results and in particular improve several recent estimates of J. Cilleruelo and M.Z. Garaev on exponential congruences and on cardinalities of products of short intervals, some double character sum estimates of J. Friedlander and H. Iwaniec and some results of M.-C. Chang and A.A. Karatsuba on character sums twisted with the divisor function.

Distance sets of well-distributed planar sets for polygonal norms

LetX be a two-dimensional normed space, and letBX be the unit ball inX. We discuss the question of how large the set of extremal points ofBX may be ifX contains a well-distributed set whose distance set Δ satisfies the estimate |Δ∩[0,N]|≤CN3/2-ε. We also give a necessary and sufficient condition for the existence of a well-distributed set with |Δ∩[0,N]|≤CN.

ARITHMETIC PROPERTIES OF INTEGERS WITH MISSING DIGITS: DISTRIBUTION IN RESIDUE CLASSES

We show that numbers with missing digits are in average well-distributed in residue classes mod m where averaging is taken over m.

Linear Complexity of the Discrete Logarithm

We obtain new lower bounds on the linear complexity of several consecutive values of the discrete logarithm modulo a prime p. These bounds generalize and improve several previous results.

On the consecutive powers of a primitive root : gaps and exponential sums

For a primitive root g modulo a prime p ≥1 we obtain upper bounds on the gaps between the residues modulo p of the N consecutive powers ag n , n =1,…, N , which is uniform over all integers a with gcd ( a , p )=1.

On the Number of Prime Facttors of Integers Characterized by Digit Properties

The number of distinct prime factors of integers with missing digits is considered, and both the normal order and large values of the ω function over sets of this type are studied. A conjecture of Mauduit and Sárközy, on large values of the Ω function over integers whose sum of digits is fixed, is also proved.

On divergence of trigonometric Fourier series everywhere

Abstract We prove that if a positive sequence ψ(m) satisfy the condition ψ(m) = o ( √log m/ √log log m) (m → ∞), then there exists a function f e L[−π,π] such that lim supSm(f,x)/ψ(m) = ∞ Ax e [−π,π], m→∞ where Sm(f,x) is the m-th partial Fourier sum of f.

Breaking the k 2 barrier for explicit RIP matrices

We give a new explicit construction of n x N matrices satisfying the Restricted Isometry Property (RIP). Namely, for some ε>0, large k and k2-ε ≤ N ≤ k2+ε, we construct RIP matrices of order k with n=O(k2-ε). This overcomes the natural barrier n >> k2 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.