2022/07/27 by Shunsuke Usuki, Usuki, Shunsuke
Mathematics · #11J13 #37A17 #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2207.13462
openalex publication_date 2022/07/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that, for any 0<γ<1/2, any (α,β)∈ℝ2 except on a set with Hausdorff dimension about √γ, any small 0<ε<1 and any large N∈ℕ, the number of integers n∈[1,N] such that n⟨ nα⟩⟨ nβ⟩<ε is greater than γεlog N up to a uniform constant. This can be seen as a quantitative result on the fact that the exceptional set to Littlewood's conjecture has Hausdorff dimension zero, obtained by M. Einsiedler, A. Katok and E. Lindenstrauss in 2000's. For the proof, we study the behavior of the empirical measures with respect to the diagonal action on \rmSL(3,ℝ)/\rmSL(3,ℤ) and show that we can obtain a quantitative result on Littlewood's conjecture for (α,β) if the corresponding empirical measures are well-behaved. We also estimate Hausdorff dimension of the exceptional set to be small.