Author:
Björklund Michael,Gorodnik Alexander
Abstract
AbstractWe consider the problem of counting lattice points contained in domains in $$\mathbb {R}^d$$
R
d
defined by products of linear forms. For $$d \ge 9$$
d
≥
9
we show that the normalized discrepancies in these counting problems satisfy non-degenerate Central Limit Theorems with respect to the unique $${\text {SL}}_d(\mathbb {R})$$
SL
d
(
R
)
-invariant probability measure on the space of unimodular lattices in $$\mathbb {R}^d$$
R
d
. We also study more refined versions pertaining to “spiraling of approximations”. Our techniques are dynamical in nature and exploit effective exponential mixing of all orders for actions of diagonalizable subgroups on spaces of unimodular lattices.
Funder
Chalmers University of Technology
Publisher
Springer Science and Business Media LLC
Subject
General Physics and Astronomy,General Mathematics
Reference28 articles.
1. Athreya, J., Ghosh, A., Tseng, J.: Spiraling of approximations and spherical averages of Siegel transforms. J. Lond. Math. Soc. 91(2), 383–404 (2015)
2. Beck, J.: Randomness in Lattice Point Problems. Combinatorics, Graph Theory, Algorithms and Applications. Discrete Math. 229(1–3), 29–55 (2001)
3. Beck, J.: Probabilistic Diophantine Approximation. Randomness in Lattice Point Counting. Springer Monographs in Mathematics. Springer, Cham (2014)
4. Björklund, M., Einsiedler, M., Gorodnik, A.: Quantitative multiple mixing. J. Eur. Math. Soc. 22(5), 1475–1529 (2020)
5. Björklund, M., Gorodnik, A.: Central limit theorems in the geometry of numbers. Electron. Res. Announc. Math. Sci. 24, 110–122 (2017)
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