Optimal Diophantine exponents for SL(n)

• Published in: Advances in mathematics (New York. 1965) Vol. 443; p. 109613
• Main Authors: Jana, Subhajit, Kamber, Amitay
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01-05-2024
• Subjects: Automorphic spectrum; Density hypothesis; Diophantine approximation; Automorphic spectrum; 11F72; Diophantine approximation; Density hypothesis; 11J13
• ISSN: 0001-8708; 1090-2082
• DOI: 10.1016/j.aim.2024.109613

The Diophantine exponent of an action of a group on a homogeneous space, as defined by Ghosh, Gorodnik, and Nevo, quantifies the complexity of approximating the points of the homogeneous space by the points on an orbit of the group. We show that the Diophantine exponent of the SLn(Z[1/p])-action on the generalized upper half-space SLn(R)/SOn(R), lies in [1,1+O(1/n)], substantially improving upon Ghosh–Gorodnik–Nevo's method which gives the above range to be [1,n−1]. We also show that the exponent is optimal, i.e. equals one, under the assumption of Sarnak's density hypothesis. The result, in particular, shows that the optimality of Diophantine exponents can be obtained even when the temperedness of the underlying representations, the crucial assumption in Ghosh–Gorodnik–Nevo's work, is not satisfied. The proof uses the spectral decomposition of the homogeneous space and bounds on the local L2-norms of the Eisenstein series.