Kloosterman Sums, Disjointness, and Equidistribution

Einsiedler, Manfred; Luethi, Manuel

doi:10.1007/978-3-319-74908-2_10

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“…These measures equidistribute on T 2 as q → ∞ [BK02, Zha96] (see also [EL18]), so that lim q→∞ µ q (B) = vol(B) for every continuity set B ⊂ T 2 , (1.2)…”

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confidence: 99%

Distributing Points on the Torus via Modular Inverses

Humphries

2020

Preprint

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We study various statistics regarding the distribution of the pointsas q tends to infinity. Due to nontrivial bounds for Kloosterman sums, it is known that these points equidistribute on the torus. We prove refinements of this result, including bounds for the discrepancy, small scale equidistribution, bounds for the covering exponent associated to these points, sparse equidistribution, and mixing. A key tool in several of the proofs is an auxiliary result that gives an asymptotic for the variance of the number of points in a random ball on the torus of arbitrarily small size, which is asymptotically identical to such a variance associated to randomly distributed points on the torus.

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“…These measures equidistribute on T 2 as q → ∞ [BK02, Zha96] (see also [EL18]), so that lim q→∞ µ q (B) = vol(B) for every continuity set B ⊂ T 2 , (1.2)…”

mentioning

confidence: 99%

Distributing Points on the Torus via Modular Inverses

Humphries

2020

Preprint

View full text Add to dashboard Cite

show abstract

Distributing Points On The Torus Via Modular Inverses

Humphries

2021

The Quarterly Journal of Mathematics

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We study various statistics regarding the distribution of the points $$\begin{equation*} \left\{\left(\frac{d}{q},\frac{\overline{d}}{q}\right) \in \mathbb{T}^2 : d \in (\mathbb{Z}/q\mathbb{Z})^{\times}\right\} \end{equation*}$$ as q tends to infinity. Due to non-trivial bounds for Kloosterman sums, it is known that these points equidistribute on the torus. We prove refinements of this result, including bounds for the discrepancy, small-scale equidistribution, bounds for the covering exponent associated with these points, sparse equidistribution, and mixing.

show abstract

Kloosterman Sums, Disjointness, and Equidistribution

Cited by 2 publications

References 27 publications

Distributing Points on the Torus via Modular Inverses

Distributing Points on the Torus via Modular Inverses

Distributing Points On The Torus Via Modular Inverses

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