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We give a new explicit construction of n × N matrices satisfying the Restricted Isometry Property (RIP). Namely, for some ε > 0, large N and any n satisfying N 1−ε ≤ n ≤ N , we construct RIP matrices of order k ≥ n 1/2+ε and constant δ −ε. This overcomes the natural barrier k = O(n 1/2) for proofs based on small coherence, which are used in all previous(More)
We give a new explicit construction of n × N matrices satisfying the Restricted Isometry Property (RIP). Namely, for some ε > 0, large k and k 2−ε ≤ N ≤ k 2+ε , we construct RIP matrices of order k with n = O(k 2−ε). This overcomes the natural barrier n k 2 for proofs based on small coherence, which are used in all previous explicit constructions of RIP(More)
We investigate the distribution of n − M (n) where M (n) = max { |a − b| : 1 ≤ a, b ≤ n − 1 and ab ≡ 1 (mod n)}. Exponential sums provide a natural tool for obtaining upper bounds on this quantity. Here we use results about the distribution of integers with a divisor in a given interval to obtain lower bounds on n − M (n). We also present some heuristic(More)
We show that the equation φ(a) = σ(b) has infinitely many solutions, where φ is Euler's totient function and σ is the sum-of-divisors function. This proves a 50-year old conjecture of Erd˝ os. Moreover, we show that there are infinitely many integers n such that φ(a) = n and σ(b) = n each have more than n c solutions, for some c > 0. The proofs rely on the(More)
A replication of an American study of body shape preference was conducted in a group of 218 Arab students attending the American University in Cairo, Egypt. Arab female students felt their ideal shape to be significantly thinner than their current shape, while male students did not. Hence the appraisal of body shape shows gender differences in Egypt(More)
We enhance the efficient congruencing method for estimating Vinogradov's integral for moments of order 2s, with 1 s k 2 − 1. In this way, we prove the main conjecture for such even moments when 1 s 1 4 (k + 1) 2 , showing that the moments exhibit strongly diagonal behaviour in this range. There are improvements also for larger values of s, these finding(More)
An old question of Erd˝ os asks if there exists, for each number N , a finite set S of integers greater than N and residue classes r(n) (mod n) for n ∈ S whose union is Z. We prove that if n∈S 1/n is bounded for such a covering of the integers, then the least member of S is also bounded, thus confirming a conjecture of Erd˝ os and Selfridge. We also prove a(More)