Quantitative Version of the Oppenheim Conjecture for Inhomogeneous Quadratic Forms
نویسندگان
چکیده
A quantitative version of the Oppenheim conjecture for inhomogeneous quadratic forms is proved. We also give an application to eigenvalue spacing on flat 2-tori with Aharonov-Bohm flux.
منابع مشابه
Pair Correlation Densities of Inhomogeneous Quadratic Forms
Under explicit diophantine conditions on (α, β) ∈ R2, we prove that the local two-point correlations of the sequence given by the values (m − α)2 + (n−β)2, with (m,n) ∈ Z2, 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 Oppen...
متن کاملValues of Indefinite Quadratic Forms at Integral Points and Flows on Spaces of Lattices
This mostly expository paper centers on recently proved conjectures in two areas: A) A conjecture of A. Oppenheim on the values of real indefinite quadratic forms at integral points. B) Conjectures of Dani, Raghunathan, and Margulis on closures of orbits in spaces of lattices such as SLn(R)/SLn(Z) . At first sight, A) belongs to analytic number theory and B) belongs to ergodic and Lie theory, a...
متن کاملUnipotent Flows and Applications
We should think of the coefficients aij of Q as real numbers (not necessarily rational or integer). One can still ask what will happen if one substitutes integers for the xi. It is easy to see that if Q is a multiple of a form with rational coefficients, then the set of values Q(Z) is a discrete subset of R. Much deeper is the following conjecture: Conjecture 1.1 (Oppenheim, 1929). Suppose Q is...
متن کاملSyllabus and Reading List for Eskin-kleinbock Course
1. General introduction, Birkhoff’s Ergodic Theorem vs. Ratner’s Theorems on unipotent flows; measure classification implies classification of orbit closures; uniform convergence and the theorem of Dani-Margulis; the statement of the Oppenheim Conjecture. 2. The case of SL(2, R) (the mixing argument). We will be loosely following Ratner’s paper [18]. 3. The classification of invariant measures ...
متن کاملOppenheim Conjecture for Pairs Consisting of a Linear Form and a Quadratic Form
Let Q be a nondegenerate quadratic form, and L is a nonzero linear form of dimension d > 3. As a generalization of the Oppenheim conjecture, we prove that the set {(Q(x), L(x)) : x ∈ Z} is dense in R provided that Q and L satisfy some natural conditions. The proof uses dynamics on homogeneous spaces of Lie groups.
متن کامل