No abstract

Abstract. Motivated by the study of billiards in polygons, we prove fine results for the distribution of gaps of directions of saddle connections on translation surfaces. As an application we prove that for almost every holomorphic differential ω on a Riemann surface of genus g ≥ 2 the smallest gap between saddle connection directions of length at most a fixed length decays faster than quadratically in the length. We also characterize the exceptional set: the decay rate is not faster than quadratic if and only if ω is a lattice surface.

We give an explicit formula for the limiting gap distribution of slopes of saddle connections on the golden L, or any translation surface in its SL(2, R)-orbit, in particular the double pentagon. This is the first explicit computation of the distribution of gaps for a flat surface that is not a torus cover. J.S.A.

We show that Sarnak's conjecture on Mobius disjointness holds for interval exchange transformations on three intervals (3-IETs) that satisfy a mild diophantine condition.

Let N represent the positive integers and N 0 the non-negative integers. If b ∈ N and Γ is a multiplicatively closed subset of Z b = Z/bZ, then the set H Γ = {x ∈ N | x + bZ ∈ Γ } ∪ {1} is a multiplicative submonoid of N known as a congruence monoid. An arithmetical congruence monoid (or ACM) is a congruence monoid where Γ = {a} consists of a single element. If H Γ is an ACM, then we represent it with the notation M (a, b) = (a + bN 0) ∪ {1}, where a, b ∈ N and a 2 ≡ a (mod b). A classical 1954 result of James and Niven implies that the only ACM which admits unique factorization of elements into products of irreducibles is M (1, 2) = M (3, 2). In this paper, we examine further factorization properties of ACMs. We find necessary and sufficient conditions for an ACM M (a, b) to be half-factorial (i.e., lengths of irreducible factorizations of an element remain constant) and further determine conditions for M (a, b) to have finite elasticity. When the elasticity of M (a, b) is finite, we produce a formula to compute it. Among our remaining results, we show that the elasticity of an ACM M (a, b) may not be accepted and show that if an ACM M (a, b) has infinite elasticity, then it is not fully elastic.