The Distribution of Lattice Points in Elliptic Annuli

Abstract

We study the distribution of the number of lattice points lying in thin elliptical annuli. It has been conjectured by Bleher and Lebowitz that, if the width of the annuli tend to zero while their area tends to infinity, then the distribution of this number, normalized to have zero mean and unit variance, is Gaussian. This has been proved by Hughes and Rudnick for circular annuli whose width shrink to zero sufficiently slowly. We prove this conjecture for ellipses whose aspect ratio is transcendental and strongly Diophantine, also assuming the width shrinks slowly to zero.

Distribution of Integer Lattice Points in a Ball Centred at a Diophantine Point

“…For α = 0, more detailed information about the distribution of S was obtained in [7]: under the condition that ρ → 0 and ρ T −δ for all δ > 0, it was shown that S has a Gaussian distribution. A similar result was established in [14] for "strongly" Diophantine rectangular lattices.…”

“…On the other hand, Theorem 1.4 hinges on the asymptotic formula (14). We reiterate that Proposition 2.3 is the only ingredient of our proof that necessitates the Diophantine properties of α.…”

“…In the present paper, however, the choice of M is virtually unrestricted (see Theorem 2.2), the unsmoothing argument is quite elementary, and a very crude estimate for the close pairs suffices (see Lemma 2.8). The explanation of this freedom of choice of M lies with the asymptotics (14) of the function R α (N ) for Diophantine αs. A naive upper bound for R α (N ), obtained by estimating r α ( p) via the number of lattice points on the sphere of radius √ p, would not be sufficient.…”

Limit theorems for toral translations

“…[15,78,86,96,105,109,110,121,122] and references wherein). In fact, one can also consider the varying shapes RD R which includes both the Poisson regime where Vol(RD R ) does not grow (see [18] and references wherein) and the intermediate regime where Vol(RD R ) grows but at the rate slower than R d (see [54,127]).…”

“…both the Poisson regime where Vol(RD R ) does not grow (see [18] and references wherein) and the intermediate regime where Vol(RD R ) grows but at the rate slower than R d (see [54,127]).…”