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We show that, for almost all x, the interval (x, x+(\log x)^{2.1}] contains products of exactly two primes. This improves on a work of the second author that had 3.51 in place of 2.1. To obtain this improvement, we prove a new type II estimate. One of the new innovations is to use Heath-Brown's mean value theorem for sparse Dirichlet polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we present a variant of the Balog-Szemerédi-Gowers theorem for the Vinogradov system. We then use our result to deduce a higher degree analogue of the sum-product phenomenon. [ABSTRACT FROM AUTHOR]

We consider Riemannian symmetric spaces X of noncompact-type with rank one, which accommodates all hyperbolic spaces. We characterize the eigenfunctions of the Laplace-Beltrami operator on X with arbitrary complex eigenvalues through an asymptotic version of the ball mean value property as the radius of the ball tends to infinity. [ABSTRACT FROM AUTHOR]

We develop a repeated efficient differencing procedure for estimating mean values of certain multidimensional exponential sums over smooth numbers. As a consequence, we obtain asymptotic lower bounds for the number of linear spaces of fixed dimension up to a given height lying on the hypersurface defined by an additive equation. [ABSTRACT FROM AUTHOR]