Abstract. We obtain an essential spectral gap for n-dimensional convex co-compact hyperbolic manifolds with the dimension δ of the limit set close to n−1 2 . The size of the gap is expressed using the additive energy of stereographic projections of the limit set. This additive energy can in turn be estimated in terms of the constants in Ahlfors-David regularity of the limit set. Our proofs use new microlocal methods, in particular a notion of a fractal uncertainty principle.In this paper we study essential spectral gaps for convex co-compact hyperbolic quotients M = Γ\H n . To formulate our result in the simplest setting, consider n = 2 and take the Selberg zeta function [Bo07, (10.1)] ), where δ ∈ [0, 1] is the dimension of the limit set of the group (see (5.2)). Therefore Z M has finitely many singularities in {Im λ > − max(0,, Naud [Na] obtained the stronger statement that Z M has finitely many singularities in {Im λ > −β} for some β strictly greater than 1 2 − δ. Naud's result, generalized to higher dimensional quotients by Stoyanov [St11], is based on the method of Dolgopyat [Do] and does not specify the size of the improvement. Our first result in particular gives explicit estimates on the value of β when δ = 1 2 : Theorem 1. Let M = Γ\H 2 be a convex co-compact hyperbolic surface. Then for each ε > 0, the function Z M has finitely many singularities in {Im λ > −β + ε}, where β = 3 8 1 2 − δ + β E 16 , β E := δ exp − K(1 − δ) −28 (1 + log 14 C) .

A bipartite graph G is semi-algebraic in R d if its vertices are represented by point sets P, Q ⊂ R d and its edges are defined as pairs of points (p, q) ∈ P × Q that satisfy a Boolean combination of a fixed number of polynomial equations and inequalities in 2d coordinates. We show that for fixed k, the maximum number of edges in a K k,kfree semi-algebraic bipartite graph G = (P, Q, E) in R 2 with |P | = m and |Q| = n is at most O((mn) 2/3 + m + n), and this bound is tight. In dimensions d ≥ 3, we show that all such semi-algebraic graphs have at most C (mn)where here ε is an arbitrarily small constant and C = C(d, k, t, ε). This result is a farreaching generalization of the classical Szemerédi-Trotter incidence theorem. The proof combines tools from several fields: VC-dimension and shatter functions, polynomial partitioning, and Hilbert polynomials.We also present various applications of our theorem. For example, a general pointvariety incidence bound in R d , an improved bound for a d-dimensional variant of the Erdős unit distances problem, and more.

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We show that a set of n algebraic plane curves of constant maximum degree can be cut into O(n 3/2 polylog n) Jordan arcs, so that each pair of arcs intersect at most once, i.e., they form a collection of pseudo-segments. This extends a similar (and slightly better) bound for pseudo-circles due to Marcus and Tardos. Our result is based on a technique of Ellenberg, Solymosi and Zahl that transforms arrangements of plane curves into arrangements of space curves, so that lenses (pairs of subarcs of the curves that intersect at least twice) become vertical depth cycles. We then apply a variant of a technique of Aronov and Sharir to eliminate these depth cycles by making a small number of cuts, which corresponds to a small number of cuts to the original planar arrangement of curves. After these cuts have been performed, the resulting curves form a collection of pseudo-segments.Our cutting bound leads to new incidence bounds between points and constantdegree algebraic curves. The conditions for these incidence bounds are slightly stricter than those for the current best-known bound of Pach and Sharir; for our result to hold, the curves must be algebraic and of bounded maximum degree, while Pach and Sharir's bound only imposes weaker, purely topological constraints on the curves. However, when our conditions hold, the new bounds are superior for almost all ranges of parameters. We also obtain new bounds on the complexity of a single level in an arrangement of constant-degree algebraic curves, and a new bound on the complexity of many marked faces in an arrangement of such curves.

We study the structure of collections of algebraic curves in three dimensions that have many curve-curve incidences. In particular, let k be a field and let L be a collection of n space curves in k 3 , with n < < (char(k)) 2 or char(k) = 0. Then either A) there are at most O(n 3/2 ) points in k 3 hit by at least two curves, or B) at least Ω(n 1/2 ) curves from L must lie on a bounded-degree surface, and many of the curves must form two "rulings" of this surface.We also develop several new tools including a generalization of the classical flecnode polynomial of Salmon and new algebraic techniques for dealing with this generalized flecnode polynomial.