Geodesic
Statistics of lattice points in thin annuli for generic lattices
Documenta mathematica, Tome 11 (2006), pp. 1-23.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: We study the statistical properties of the counting function of lattice points inside thin annuli. By a conjecture of Bleher and Lebowitz, if the width shrinks to zero, but the area converges to infinity, the distribution converges to the Gaussian distribution. If the width shrinks slowly to zero, the conjecture was proven by Hughes and Rudnick for the standard lattice, and in our previous paper for generic rectangular lattices. We prove this conjecture for arbitrary lattices satisfying some generic Diophantine properties, again assuming the width of the annuli shrinks slowly to zero. One of the obstacles of applying the technique of Hughes-Rudnick on this problem is the existence of so-called close pairs of lattice points. In order to overcome this difficulty, we bound the rate of occurence of this phenomenon by extending some of the work of Eskin-Margulis-Mozes on the quantitative Openheim conjecture.
Classification : 11H06, 11J25
Keywords: lattice, counting function, circle, ellipse, annulus, two-dimensional torus, Gaussian distribution, Diophantine approximation 
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     author = {Wigman, Igor},
     title = {Statistics of lattice points in thin annuli for generic lattices},
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     year = {2006},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2006__11__a17/}
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Wigman, Igor. Statistics of lattice points in thin annuli for generic lattices. Documenta mathematica, Tome 11 (2006), pp. 1-23. http://geodesic.mathdoc.fr/item/DOCMA_2006__11__a17/