In this paper we prove results on Birkhoff and Oseledets genericity along certain curves in the space of affine lattices and in moduli spaces of translation surfaces. We also prove applications of these results to dynamical billiards, mathematical physics and number theory. In the space of affine lattices ASL 2 (R)/ASL 2 (Z), we prove that almost every point on a curve with some non-degeneracy assumptions is Birkhoff generic for the geodesic flow. This implies almost everywhere genericity for some curves in the locus of branched covers of the torus inside the stratum H(1, 1) of translation surfaces. For these curves (and more in general curves which are well-approximated by horocycle arcs and satisfy almost everywhere Birkhoff genericity) we also prove that almost every point is Oseledets generic for the Kontsevitch-Zorich cocycle, generalizing a recent result by Chaika and Eskin. As applications, we first consider a class of pseudo-integrable billiards, billiards in ellipses with barriers, which was recently explored by Dragović and Radnović, and prove that for almost every parameter, the billiard flow is uniquely ergodic within the region of phase space in which it is trapped. We then consider any periodic array of Eaton retroreflector lenses, placed on vertices of a lattice, and prove that in almost every direction light rays are each confined to a band of finite width. This generalizes a phenomenon recently discovered by Frączek and Schmoll which could so far only be proved for random periodic configurations. Finally, a result on the gap distribution of fractional parts of the sequence of square roots of positive integers, which extends previous work by Elkies and McMullen, is also obtained.

Abstract. Consider G = SL d (R) and Γ = SL d (Z). It was recently shown by the second-named author [21] that for some diagonal subgroups {g t } ⊂ G and unipotent subgroups U ⊂ G, g ttrajectories of almost all points on all U -orbits on G/Γ are equidistributed with respect to continuous compactly supported functions ϕ on G/Γ. In this paper we strengthen this result in two directions: by exhibiting an error rate of equidistribution when ϕ is smooth and compactly supported, and by proving equidistribution with respect to certain unbounded functions, namely Siegel transforms of Riemann integrable functions on R d . For the first part we use a method based on effective double equidistribution of g ttranslates of U -orbits, which generalizes the main result of [13]. The second part is based on Schmidt's results on counting of lattice points. Number-theoretic consequences involving spiraling of lattice approximations, extending recent work of Athreya, Ghosh and Tseng [1], are derived using the equidistribution result.

Let U be a horospherical subgroup of a noncompact simple Lie group H and let A be a maximal split torus in the normalizer of U . We define the expanding cone AÙ in A with respect to U and show that it can be explicitly calculated. We prove several dynamical results for translations of U -slices by elements of AÙ on finite volume homogeneous space G{Γ where G is a Lie group containing H. More precisely, we prove quantitative nonescape of mass and equidistribution of a U -slice. If H is a normal subgroup of G and the H action on G{Γ has a spectral gap, we prove effective multiple equidistribution and pointwise equidistribution with an error rate. In the paper we formulate the notion of the expanding cone and prove the dynamical results above in the more general setting where H is a semisimple Lie group without compact factors.In the appendix, joint with Rene Rühr, we prove a multiple ergodic theorem with an error rate.

Let Γ be a lattice of a semisimple Lie group L. Suppose that one parameter Ad-diagonalizable subgroup {g t } of L acts ergodically on L/Γ with respect to the probability Haar measure µ. For certain proper subgroup U of the unstable horospherical subgroup of {g t } and certain x ∈ L/Γ we show that for almost every u ∈ U the trajectory {g t ux : 0 ≤ t ≤ T } is equidistributed with respect to µ as T → ∞.

Abstract. In this note we give a detailed proof of certain results on geometry of numbers in the S-adic case. These results are well-known to experts, so the aim here is to provide a convenient reference for the people who need to use them.