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- W4311881508 abstract "Abstract We consider the problem of counting lattice points contained in domains in $$mathbb {R}^d$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> defined by products of linear forms. For $$d ge 9$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>9</mml:mn> </mml:mrow> </mml:math> we show that the normalized discrepancies in these counting problems satisfy non-degenerate Central Limit Theorems with respect to the unique $${text {SL}}_d(mathbb {R})$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:msub> <mml:mtext>SL</mml:mtext> <mml:mi>d</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> -invariant probability measure on the space of unimodular lattices in $$mathbb {R}^d$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> . 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." @default.
- W4311881508 created "2023-01-02" @default.
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- W4311881508 date "2022-12-08" @default.
- W4311881508 modified "2023-09-26" @default.
- W4311881508 title "Central limit theorems for generic lattice point counting" @default.
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- W4311881508 doi "https://doi.org/10.1007/s00029-022-00815-w" @default.
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