Gaps in √n mod 1 and ergodic theory

Small solutions to linear congruences and Hecke equidistribution

“…The estimates for Hecke operators that are used in Section 3 are already in [COU], and Sobolev norms are used elegantly in [GO]. The computations in Section 8 are inspired by the method of Elkies and McMullen [EM,.…”

Effective Ratner theorem for SL(2,R)⋉R2 and gaps in n modulo 1

“…L(Γg) is the area of the largest triangle in the family ∆ c − ,c + that is disjoint from Z 2 g. Following [8], we establish a connection between homogeneous dynamics (embodied in the function L) and number theoretic quantities (embodied in σ N and λ N ). This is achieved in Lemmas 5.2, 5.3, 5.4.…”

“…This function satisfies λ N (0) = 0 and λ N (∞) = 1, and it is left-continuous. The behaviour of λ N (t), as N → ∞, has been analyzed by Elkies and McMullen [8] and later also by Sinai [14]. It is shown in [8] that there exists a function λ ∞ (t) such that λ N (t) → λ ∞ (t) for each t. We have…”

“…The main obstruction to treating the problem of horocycle lifts with the usual techniques of ergodic theory (such as thickening) is the fact that U is not the entire unstable manifold of the flow a(y), but only a codimension 1 submanifold. Elkies and McMullen [8] used Ratner's measure classification theorem [11] to prove that the horocycle lifts equidistribute, but their method is ineffective. In [8, §3.6] they ask whether explicit error estimates can be obtained.…”

“…The key input in [8] comes from Ratner's theorem [11], which is used in [8, Thm. 2.2] to find the limiting distribution of λ N (t) and therefore cannot give a rate of convergence.…”