Pair correlation densities of inhomogeneous quadratic forms

Abstract

Under explicit diophantine conditions on (α, β) ∈ R 2 , we prove that the local two-point correlations of the sequence given by the values (m − α) 2 + (n−β) 2 , with (m, n) ∈ Z 2 , are those of a Poisson process. This partly confirms a conjecture of Berry and Tabor [2] on spectral statistics of quantized integrable systems, and also establishes a particular case of the quantitative version of the Oppenheim conjecture for inhomogeneous quadratic forms of signature (2,2). The proof uses theta sums and Ratner's classification of measures invariant under unipotent flows.

Mean square value of exponential sums related to the representation of integers as sums of squares

“…The space Γ k \G k is noncompact, and Θ f Θ g is in fact unbounded. The following proposition controls the behaviour in the cusps [8,Prop. 4.10].…”

“…Following the proof of [8,Prop. 5.4] one shows that the family of probability measures { t : t ≥ 0} is relatively compact, that is, every sequence contains a subsequence which weakly converges to a probability measure on Γ \G k .…”

“…If f, g ∈ S(R + ), the function Θ f Θ g can be realized as a smooth function on a homogeneous space Γ k \G k of finite measure; see Sections 3 and 4 in [8] for more details. Here G k is the semi-direct product group G k = SL(2, R) R 2k with multiplication law…”

“…The ergodic theory of unipotent flows has proved to be a very useful tool in understanding the distribution of values of quadratic forms at integer argument (see [6]- [9] and references therein). In the present paper we use the approach developed in [8], [9] to calculate the mean square value of the exponential sums . .…”

Spherical averages in the space of marked lattices

“…If the space X is homogeneous, then the weak convergence of the spherical average P t can be proved for all x 0 , with a complete classification of all limit measures, by means of measure rigidity techniques that are based on Ratner's measure classification theorem for subgroups generated by unipotent elements [22]. There is now a large body of literature on this topic, see for instance [7,8,16,17,19,24]. In some settings, spherical equidistribution may also be deduced directly from the mixing property of R [7].…”