Erratum

Igor Wigman

Received: November 27, 2007

Communicated by Friedrich Götze

Abstract. This is an erratum concerning the article ”Statistics of Lattice Points in Thin Annuli for Generic Lattices”, Documenta Math. 11 (2005), 1–23

2000 Mathematics Subject Classification: Primary: 11H06, Secondary: 11J25 Keywords and Phrases: Lattice, Counting Function, Circle, Ellipse, Annulus, Two-Dimensional Torus, Gaussian Distribution, Diophantine approximation

1. The correct statement of Lemma 4.1 is: for every a > 0,

N_{Λ}(t) = \frac{π}{d} t^{2} − \frac{\sqrt{t}}{dπ} ∑_{\begin{matrix}k⃗∈Λ^{∗}\\{0\} \\ |k⃗|≤\sqrt{N} \end{matrix}}\frac{cos (2πt|k⃗| + \frac{π}{4} )}{|k⃗|^{\frac{3}{2}}} + O(N^{a}) + O(\frac{t^{3}}{δ_{Λ}(t^{2})\sqrt{N}} ),

provided that

δ_{Λ}(y) < 1  (1)

for all y > 0.

The proof of this lemma proceeds exactly as the proof of the correct statement corresponding to lemma 4.1 of [W] (see the erratum to [W]).

2. In the course of ”unsmoothing” (that is, the proof of lemma 4.2), we invoke lemma 4.1 with δ_{Λ}(y) = \frac{c_{1}}{y^{K_{0}}} and δ_{Λ^{∗}}(y) = \frac{c_{2}}{y^{K_{0}}} , where c_{1}, c_{2} are constants. We choose N = T^{H}, with

H := 8 + 4K_{0},

and proceed as in the original text.

Acknowledgement The author wishes to thank D. R. Heath-Brown for finding the mistake as well as other useful comments.

References

[W] Wigman, Igor The distributions of lattice points in elliptic annuli. Q. J. Math. 57 (2006), no. 3, 395–423.

Igor Wigman School of Mathematical Sciences Tel Aviv University Tel Aviv 69078 Israel igorv@post.tau.ac.il