Small solutions to linear congruences and Hecke equidistribution

Strömbergsson, Andreas; Venkatesh, Akshay

doi:10.4064/aa118-1-4

Cited by 23 publications

(26 citation statements)

References 16 publications

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“…and hence by (17), Φ 0 (ξ, w, z) equals (6/π 2 ) times the Lebesgue measure of the set of those v ∈ (0, 1) for which the lattice spanned by a 1 = (ξ, z + w) and a 2 = (ξv, v(z + w) + ξ −1 ) (23) has no point in R (z) ξ . Now for any given v ∈ (0, 1) we have ℓa 1 / ∈ R (z) ξ for all ℓ ∈ Z, by considering the first coordinate.…”

Section: (Z)mentioning

confidence: 99%

Kinetic transport in the two-dimensional periodic Lorentz gas

Marklof

Strömbergsson

2008

Nonlinearity

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Abstract. The periodic Lorentz gas describes an ensemble of non-interacting point particles in a periodic array of spherical scatterers. We have recently shown that, in the limit of small scatterer density (Boltzmann-Grad limit), the macroscopic dynamics converges to a stochastic process, whose kinetic transport equation is not the linear Boltzmann equation-in contrast to the Lorentz gas with a disordered scatterer configuration. The present paper focuses on the two-dimensional set-up, and reports an explicit, elementary formula for the collision kernel of the transport equation.One of the central challenges in kinetic theory is the derivation of macroscopic evolution equations-describing for example the dynamics of an electron gas-from the underlying fundamental microscopic laws of classical or quantum mechanics. An iconic mathematical model in this research area is the Lorentz gas [15], which describes an ensemble of non-interacting point particles in an infinite array of spherical scatterers. In the case of a disordered scatterer configuration, well known results by Gallavotti [12], Spohn [22], and Boldrighini, Bunimovich and Sinai [5] show that the time evolution of a macroscopic particle cloud is governed, in the limit of small scatterer density (Boltzmann-Grad limit), by the linear Boltzmann equation. We have recently proved an analogous statement for a periodic configuration of scatterers [16], [17]. In this case the linear Boltzmann equation fails (cf. also Golse [13]), and the random flight process that emerges in the Boltzmann-Grad limit is substantially more complicated.In the present paper we focus on the two-dimensional case, and derive explicit formulae for the collision kernels of the limiting random flight process. These include information not only of the velocity before and after the collision (as in the case of the linear Boltzmann equation), but also the path length until the next hit and the velocity thereafter. Our formulae thus generalize those for the limiting distributions of the free path length found by Dahlqvist [10] Our results on the Boltzmann-Grad limit complement classical studies in ergodic theory, where the scatterer size remains fixed. Bunimovich and Sinai [7] showed that the dynamics of the two-dimensional periodic Lorentz gas is diffusive in the limit of large times, and satisfies a central limit theorem. They assumed that the periodic Lorentz gas has a finite horizon, i.e., the scatterers are configured in such a way that the path length between consecutive collisions is bounded. The corresponding result for infinite horizon has recently been established by Szasz and Varju [21] following initial work by Bleher [2]. See also the recent papers by Dolgopyat, Szasz and Varju [11], and Melbourne and Nicol [19], [20] for related studies of statistical properties of the two-dimensional periodic Lorentz gas. The case of higher dimensions is still open, even for models with finite horizon, cf. Chernov [9], and Balint and Toth [1]. Scaling limits that are intermediate between the ...

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Section: (Z)mentioning

confidence: 99%

Kinetic transport in the two-dimensional periodic Lorentz gas

Marklof

Strömbergsson

2008

Nonlinearity

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“…We give tail estimates for F 0 .0; / and F 0;0 .0; / for general dimension d in [25]. In the special case d D 2, explicit formulas for F 0 .r; / and F 0;0 .r; / were given in [36], where these limit functions came up in a different set of problems. Specifically, absolutely continuous with respect to Lebesgue measure.…”

Section: The Number Of Spheres In a Random Directionmentioning

confidence: 99%

The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems

2010

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The periodic Lorentz gas describes the dynamics of a point particle in a periodic array of spherical scatterers, and is one of the fundamental models for chaotic diffusion. In the present paper we investigate the Boltzmann-Grad limit, where the radius of each scatterer tends to zero, and prove the existence of a limiting distribution for the free path length. We also discuss related problems, such as the statistical distribution of directions of lattice points that are visible from a fixed position.

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“…Basically we need control on the L 1 -norm of "1 + ε" derivatives of ν; in order to avoid a technical overhead we formulate the bound using a crude interpolation between the Sobolev norms ν W 1,1 and ν W 2,1 (cf., e.g., [48,Sec. 2]).…”

Section: Smoothed Ergodic Averagesmentioning

confidence: 99%

“…Now choose ℓ ≥ 1 so that q ℓ−1 ≤ (cX) (44) follows by adding (45), (48) and the bound n>X σ(n)n −2 ≪ ε X ε−1 , replacing ε by 1 3 ε, and using (cX)…”

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confidence: 99%