For each positive integer n, construct a square graph with boundary Γ = (V, VB, E) as follows. V is the set of vertices in the graph and consists of the integer lattice points (x, y) where 0 ≤ x ≤… (More)

Using the polynomial method of Dvir [5], we establish optimal estimates for Kakeya sets and Kakeya maximal functions associated to algebraic varieties W over finite fields F . For instance, given an… (More)

Abstract. We prove old and new L bounds for the quartile operator, a Walsh model of the bilinear Hilbert transform, uniformly in the parameter that models degeneration of the bilinear Hilbert… (More)

A (d, k) set is a subset of Rd containing a translate of every k-dimensional plane. Bourgain showed that for k ≥ kcr(d), where kcr(d) solves 2kcr−1 + kcr = d, every (d, k) set has positive Lebesgue… (More)

We prove almost sharp mixed-norm estimates for the X-ray transform restricted to lines whose directions lie on certain well-curved two dimensional manifolds.

We study some discrete and continuous variants of the following problem of Erd˝ os: given a finite subset P of R 2 or R 3 , what is the maximum number of pairs (p1, p2) with p1, p2 ∈ P and |p1 −p2| =… (More)

Abstract. For any dynamical system, we show that higher variation-norms for the sequence of ergodic bilinear averages of two functions satisfy a large range of bilinear Lp estimates. It follows that,… (More)

A (d, k) set is a subset of R containing a translate of every k-dimensional disc of diameter 1. We show that if (1 + √ 2)k−1 + k > d and k ≥ 2, then every (d, k) set has positive Lebesgue measure.… (More)