Abstract. For a sequence of integers {a(x)} x≥1 we show that the distribution of the pair correlations of the fractional parts of { αa(x) } x≥1 is asymptotically Poissonian for almost all α if the additive energy of truncations of the sequence has a power savings improvement over the trivial estimate. Furthermore, we give an estimate for the Hausdorff dimension of the exceptional set as a function of the density of the sequence and the power savings in the energy estimate. A consequence of these results is that the Hausdorff dimension of the set of α such that { αx d } fails to have Poissonian pair correlation is at most d+2 d+3 < 1. This strengthens a result of Rudnick and Sarnak which states that the exceptional set has zero Lebesgue measure. On the other hand, classical examples imply that the exceptional set has Hausdorff dimension at least 2 d+1 .

Abstract. We give a simple proof of a well-known theorem of Gál and of the recent related results of Aistleitner, Berkes and Seip [1] regarding the size of GCD sums. In fact, our method obtains the asymptotically sharp constant in Gál's theorem, which is new. Our approach also gives a transparent explanation of the relationship between the maximal size of the Riemann zeta function on vertical lines and bounds on GCD sums; a point which was previously unclear. Furthermore we obtain sharp bounds on the spectral norm of GCD matrices which settles a question raised in [2]. We use bounds for the spectral norm to show that series formed out of dilates of periodic functions of bounded variation converge almost everywhere if the coefficients of the series are in L 2 (log log 1/L) γ , with γ > 2. This was previously known with γ > 4, and is known to fail for γ < 2. We also develop a sharp Carleson-Hunt-type theorem for functions of bounded variations which settles another question raised in [1]. Finally we obtain almost sure bounds for partial sums of dilates of periodic functions of bounded variations improving [1]. This implies almost sure bounds for the discrepancy of {n k x} with n k an arbitrary growing sequences of integers.

We investigate the square variation operator V 2 (which majorizes the partial sum maximal operator) on general orthonormal systems (ONS) of size N . We prove that the L 2 norm of the V 2 operator is bounded by O(ln(N )) on any ONS. This result is sharp and refines the classical Rademacher-Menshov theorem. We show that this can be improved to O( ln(N )) for the trigonometric system, which is also sharp. We show that for any choice of coefficients, this truncation of the trigonometric system can be rearranged so that the L 2 norm of the associated V 2 operator is O( ln ln(N )). We also show that for p > 2, a bounded ONS of size N can be rearranged so that the L 2 norm of the V p operator is at most O p (ln ln(N )) uniformly for all choices of coefficients. This refines Bourgain's work on Garsia's conjecture, which is equivalent to the V ∞ case. Several other results on operators of this form are also obtained. The proofs rely on combinatorial and probabilistic methods.

We improve the range of exponents for the restriction problem for the 3-d paraboloid over finite fields. The key new ingredient is a variant of the Bourgain-Katz-Tao finite field incidence theorem derived from sum-product estimates. In prime order fields, we give an explicit relationship between the exponent in this incidence theorem and restriction estimates for the paraboloid.

We prove certain endpoint restriction estimates for the paraboloid over finite fields in three and higher dimensions. Working in the bilinear setting, we are able to pass from estimates for characteristic functions to estimates for general functions while avoiding the extra logarithmic power of the field size which is introduced by the dyadic pigeonhole approach. This allows us to remove logarithmic factors from the estimates obtained by Mockenhaupt and Tao in three dimensions and those obtained by Iosevich and Koh in higher dimensions.