We prove that the number of directions contained in a set of the form A × B ⊂ AG(2, p), where p is prime, is at least |A||B|−min{|A|, |B|}+2. Here A and B are subsets of GF (p) each with at least two elements and |A||B| < p. This bound is tight for an infinite class of examples. Our main tool is the use of the Rédei polynomial with Szőnyi's extension. As an application of our main result, we obtain an upper bound on the clique number of a Paley graph, matching the current best bound obtained recently by Hanson and Petridis.
We prove that the number of directions contained in a set of the form A × B ⊂ AG(2, p), where p is prime, is at least |A||B| − min{|A|, |B|} + 2. Here A and B are subsets of GF (p) each with at least two elements and |A||B| < p. This bound is tight for an infinite class of examples. Our main tool is the use of the Rédei polynomial with Szőnyi's extension.
We use recent advances on the discretized sum-product problem to obtain new bounds on the Hausdorff dimension of planar (α, 2α)-Fursterberg sets. This provides a quantitative improvement to the 2α + ǫ bound of Héra-Shmerkin-Yavicoli. In particular, we show that every 1/2-Furstenberg set has dimension at least 1 + 1/4536.
Let H be a multiplicative subgroup of F * p of order H > p 1/4 . We show thatwhere e p (z) = exp(2πiz/p), which improves a result of Bourgain and Garaev (2009). We also obtain new estimates for double exponential sums with product nx with x ∈ H and n ∈ N for a short interval N of consecutive integers.
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