Effective equidistribution for multiplicative Diophantine approximation on lines

• 成果类型: Article
• 署名作者: Chow, Sam; Yang, Lei
• 署名单位: University of Warwick; Institute for Advanced Study - USA
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-023-01233-1
• 发表日期: 2024
• 页码: 973-1007
• 关键词: invariant-measures; unipotent flows; logarithm laws; dirichlets theorem; conjecture; CURVES; distributions; averages; numbers; version

摘要

Given any line in the plane, we strengthen the Littlewood conjecture by two logarithms for almost every point on the line, thereby generalising the fibre result of Beresnevich, Haynes, and Velani. To achieve this, we prove an effective asymptotic equidistribution result for one-parameter unipotent orbits in SL(3, R)/SL(3, Z). We also provide a complementary convergence statement, by developing the structural theory of dual Bohr sets: at the cost of a slightly stronger Diophantine assumption, this sharpens a result of Kleinbock's from 2003. Finally, we refine the theory of logarithm laws in homogeneous spaces.