Denote by $\| \cdot \|$ the euclidean norm in $\RR^k$. We prove that the local pair correlation density of the sequence $\| \vecm -\vecalf \|^k$, $\vecm\in\ZZ^k$, is that of a Poisson process, under diophantine conditions on the fixed vector $\vecalf\in\RR^k$: in dimension two, vectors $\vecalf$ of any diophantine type are admissible; in higher dimensions ($k>2$), Poisson statistics are only observed for diophantine vectors of type $κ<(k-1)/(k-2)$. Our findings support a conjecture of Berry and Tabor on the Poisson nature of spectral correlations in quantized integrable systems.
Under explicit diophantine conditions on $(α,β)\in\RR^2$, we prove that the local two-point correlations of the sequence given by the values $(m-α)^2+\break (n-β)^2$, with $(m,n)\in\ZZ^2$, 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.
We prove an analogue of the Oppenheim conjecture for a system comprising an inhomogeneous quadratic form and a linear form in $3$ variables using dynamics on the space of affine lattices.
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.
We investigate the density of integer solutions to certain binary inhomogeneous quadratic congruences and use this information to detect almost primes on a singular del Pezzo surface of degree 6.
An integral quadratic polynomial (with positive definite quadratic part) is called almost universal if it represents all but finitely many positive integers. In this paper, we provide a characterization of almost universal ternary quadratic polynomials with conductor 2.
We establish effective versions of Oppenheim's conjecture for generic inhomogeneous quadratic forms. We prove such results for fixed quadratic forms and generic shifts. Our results complement our companion paper where we considered generic forms and fixed shifts. In this paper, we use ergodic theorems and in particular we establish a strong spectral gap with effective bounds for some representations of orthogonal groups which do not possess Kazhdan's property (T).
Let $K$ be a real quadratic field. We use a symbolic coding of the action of a fundamental unit on the real $2$-torus associated to $K$ to study the family of subsets $X_t$ of norm distance $\geq t$ from the origin. As an application, we prove that inhomogeneous spectrum of $K$ contains a dense set of elements of $K$, and conclude that all isolated inhomogeneous minima lie in $K$.
We establish effective versions of Oppenheim's conjecture for generic inhomogeneous quadratic forms. We prove such results for fixed shift vectors and generic quadratic forms. When the shift is rational we prove a counting result which implies the optimal density for values of generic inhomogeneous forms. We also obtain a similar density result for fixed irrational shifts satisfying an explicit Diophantine condition. The main technical tool is a formula for the second moment of Siegel transforms on certain congruence quotients of $\operatorname{SL}_n(\mathbb{R})$ which we believe to be of inde…
Let $G=$SL$(2,R)\ltimes(R^2)^{\oplus k}$ and let $Γ$ be a congruence subgroup of SL$(2,Z)\ltimes(Z^2)^{\oplus k}$. We prove a polynomially effective asymptotic equidistribution result for special types of unipotent orbits in $Γ\backslash G$ which project to pieces of closed horocycles in SL$(2,Z)\backslash$SL$(2,R)$. As an application, we prove an effective quantitative Oppenheim type result for the quadratic form $(m_1-α)^2+(m_2-β)^2-(m_3-α)^2-(m_4-β)^2$, for $(α,β)$ of Diophantine type, following the approach by Marklof [24] using theta sums.
We give formulas for local densities of diagonal integral ternary quadratic forms at odd primes. Exponential sums and quadratic Gauss sums are used to obtain these formulas. These formulas (along with 2-adic densities and Siegel's mass formula) can be used to compute the representation numbers of certain ternary quadratic forms.
Using the Ratios Conjecture, we write down precise formulas with lower order terms for the one and the two level densities of zeros of quadratic Dirichlet $L$--functions over function fields. We denote the various terms arising as Type-$0$, Type-I and Type-II contributions. When the support of the Fourier transform of the test function is sufficiently restricted, we rigorously compute the Type-$0$ and Type-I terms and confirm that they match the conjectured answer. When the restrictions on the support are relaxed, our results suggest that Type-II contributions become important in the two level…
Small solutions to inhomogeneous and homogeneous quadratic congruences modulo prime powers
Ramakrishna Mission Vivekananda Educational and Research Institute（拉马克里希纳传教士维韦卡南达教育与研究所）；Department of Mathematics, A. P. C. Roy Government College（A.P.C.罗伊政府学院数学系）
The celebrated Smith-Minkowski-Siegel mass formula expresses the mass of a quadratic lattice (L, Q) as a product of local factors, called the local densities of (L,Q). This mass formula is an essential tool for the classification of integral quadratic lattices. In this paper, we will describe the local density formula explicitly, by constructing a smooth integral group scheme model for an appropriate orthogonal group. Our method works for any unramified finite extension of Q_2. Therefore, we give a long awaited proof for the local density formula of Conway and Sloane and discover its generaliz…
We prove effective versions of Oppenheim's conjecture for generic inhomogeneous forms in the S-arithmetic setting. We prove an effective result for fixed rational shifts and generic forms and we also prove a result where both the quadratic form and the shift are allowed to vary. In order to do so, we prove analogues of Rogers' moment formulae for $S$-arithmetic congruence quotients as well as for the space of affine lattices. We believe the latter results to be of independent interest.
Previous work by Rubinstein and Gao computed the n-level densities for families of quadratic Dirichlet L-functions for test functions where the sum of the supports of the Fourier transforms is at most 2, and showed agreement with random matrix theory predictions in this range for n < 4 but only in a restricted range for larger n. We extend these results and show agreement for n < 8, and reduce higher n to a Fourier transform identity. The proof involves adopting a new combinatorial perspective to convert all terms to a canonical form, which facilitates the comparison of the two sides.
Conner and Hurrelbrink established a method of determining the structure of the 2-Sylow subgroup of the tame kernel $K_2(\mathcal{O})$ for certain quadratic number fields. Specifically, the 4-rank for these fields was characterized in terms of positive definite binary quadratic forms. Numerical calculations led to questions concerning possible density results of the 4-rank of tame kernels. In this paper, we succeed in giving affirmative answers to these questions.
We make some corrections to Nipp's tables of positive definite integral quaternary quadratic forms. They only affect $p$-adic densities and $p$-adic Jordan splittings in the Appendix of Nipp's book. There are no errors found in Nipp's list of quaternary forms for each genus.
For certain real quadratic number fields, we prove density results concerning 4-ranks of tame kernels. We also discuss a relationship between 4-ranks of tame kernels and 4-class ranks of narrow ideal class groups. Additionally, we give a product formula for a local Hilbert symbol.
