We provide an improvement over Meshulam's bound on cap sets in F N 3 . We show that there exist universal ǫ > 0 and C > 0 so that any cap set in F N 3 has size at most C 3 N N 1+ǫ . We do this by obtaining quite strong information about the additive combinatorial properties of the large spectrum.

Abstract. Stein conjectured that the Hilbert transform in the direction of a vector field is bounded on, say, L 2 whenever v is Lipschitz. We establish a wide range of L p estimates for this operator when v is a measurable, non-vanishing, one-variable vector field in R 2 . Aside from an L 2 estimate following from a simple trick with Carleson's theorem, these estimates were unknown previously. This paper is closely related to a recent paper of the first author ([2]).

Abstract. We prove L p , p ∈ (1, ∞) estimates on the Hilbert transform along a one variable vector field acting on functions with frequency support in an annulus. Estimates when p > 2 were proved by Lacey and Li in [4]. This paper also contains key technical ingredients for a companion paper [3] with Christoph Thiele in which L p estimates are established for the full Hilbert transform.

Abstract. We completely characterize the boundedness of planar directional maximal operators on L p . More precisely, if Ω is a set of directions, we show that M Ω , the maximal operator associated to line segments in the directions Ω, is unbounded on L p , for all p < ∞, precisely when Ω admits Kakeya-type sets. In fact, we show that if Ω does not admit Kakeya sets, then Ω is a generalized lacunary set, and hence M Ω is bounded on L p , for p > 1.

Let Cn be the n-th generation in the construction of the middlehalf Cantor set. The Cartesian square Kn = Cn × Cn consists of 4 n squares of side-length 4 −n . The chance that a long needle thrown at random in the unit square will meet Kn is essentially the average length of the projections of Kn, also known as the Favard length of Kn. A classical theorem of Besicovitch implies that the Favard length of Kn tends to zero. It is still an open problem to determine its exact rate of decay. Until recently, the only explicit upper boundwas exp(−c log * n), due to Peres and Solomyak. (log * n is the number of times one needs to take log to obtain a number less than 1 starting from n). In [11] the power estimate from above was obtained. The exponent in [11] was less than 1/6 but could have been slightly improved. On the other hand, a simple estimate shows that from below we have the estimate c n . Here we apply the idea from [4], [1] to show that the estimate from below can be in fact improved to c log n n . This is in drastic difference from the case of random Cantor sets studied in [13].