An effective Ratner equidistribution result for SL(2,R)⋉R2

Abstract

Abstract. Let G = SL(2, R) ⋉ R 2 be the affine special linear group of the plane, and set Γ = SL(2, Z) ⋉ Z 2 . We prove a polynomially effective asymptotic equidistribution result for the orbits of a 1-dimensional, non-horospherical unipotent flow on Γ\G.

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

“…They have many applications beyond the world of dynamics, ranging from problems in number theory to mathematical physics. This paper is concerned with the problem of obtaining effective versions of results that build on Ratner's theorem and is inspired by recent work of Strömbergsson [15].…”

“…Let a(y) = √ y 0 0 1/ √ y , and write A + = {a(y) : y > 0}. In what follows we will use the embedding SL(2, R) → G, given by M → (M, 0), which thereby allows us to think of SL(2, R) as a subgroup of G. Strömbergsson [15] works with the unipotent flow on X generated by right multiplication by the subgroup…”

“…In [15,Thm. 1.2], effective rates of convergence are obtained for the equidistribution of such orbits under the flow a(y) as y → 0.…”

“…Let ρ : R → R 0 be a compactly supported function that has 1 + ε derivatives in L 1 . For simplicity we follow [15] and interpolate between the Sobolev norms ρ W 1,1 and ρ W 2,1 , which give the L 1 norms of first and second derivatives, respectively. This interpolation allows us to treat the case of piecewise constant functions with an ε-loss in the rate.…”

“…The constant K in this result is absolute and does not depend on η. The proof of Theorems 1.1 and 1.2 builds on the proof of [15,Thm. 1.2].…”