We give an explicit classification of translation-invariant, Lorentz-invariant continuous valuations on convex sets. We also classify the Lorentz-invariant even generalized valuations.where a valuation φ is called even (resp. odd) if φ(−K) = φ(K) (resp. φ(−K) = −φ(K)) for any K.It is easy to see that V al 0 (R n ) is spanned by the Euler characteristic, i.e. valuation which is equal to 1 on any convex compact set. By a theorem of Hadwiger [14], V al n (R n ) is spanned by the Lebesgue measure.We denote by V al(R n ) SO + (n−1,1) the subspace of SO + (n − 1, 1)-invariant valuations, and similarly for subspaces of given parity and homogeneity. Mc-Mullen's decomposition (1) immediately impliesOur first main result classifies odd SO + (n − 1, 1)-invariant valuations.Theorem 1.1. For 0 ≤ k ≤ n, k = n − 1, V al odd k (R n ) SO + (n−1,1) = 0. For k = n − 1, dim V al odd k (R n ) SO + (n−1,1) = 1, n ≥ 3 2, n = 2as claimed. Q.E.D.
We study the fluctuations in the distribution of zeros of zeta functions of a family of hyperelliptic curves defined over a fixed finite field, in the limit of large genus. According to the Riemann hypothesis for curves, the zeros all lie on a circle. Their angles are uniformly distributed, so for a curve of genus g a fixed interval I will contain asymptotically 2g|I| angles as the genus grows. We show that for the variance of number of angles in I is asymptotically (2/π 2 ) log(2g|I|) and prove a central limit theorem: the normalized fluctuations are Gaussian. These results continue to hold for shrinking intervals as long as the expected number of angles 2g|I| tends to infinity.
Let $\mathrm{SO}^+(p,q)$ denote the identity connected component of the real
orthogonal group with signature $(p,q)$. We give a complete description of the
spaces of continuous and generalized translation- and
$\mathrm{SO}^+(p,q)$-invariant valuations, generalizing Hadwiger's
classification of Euclidean isometry-invariant valuations. As a result of
independent interest, we identify within the space of translation-invariant
valuations the class of Klain-Schneider continuous valuations, which strictly
contains all continuous translation-invariant valuations. The operations of
pull-back and push-forward by a linear map extend naturally to this class.Comment: 65 pages; Some details in proofs added and minor mistakes corrected,
an appendix on wave front sets of G-invariant distributions and a section on
the linear algebra of O(p,q) added. Accepted for publication in Journal of
Functional Analysi
Abstract. We study the O(p, q)-invariant valuations classified by A. Bernig and the author. Our main result is that every such valuation is given by an O(p, q)-invariant Crofton formula. This is achieved by first obtaining a handful of explicit formulas for a few sufficiently general signatures and degrees of homogeneity, notably in the (p − 1) homogeneous case of O(p, p), yielding a Crofton formula for the centro-affine surface area when p ≡ 3 mod 4. We then exploit the functorial properties of Crofton formulas to pass to the general case. We also identify the invariant formulas explicitly for all O(p, 2)-invariant valuations. The proof relies on the exact computation of some integrals of independent interest. Those are related to Selberg's integral and to the Beta function of a matrix argument, except that the positive-definite matrices are replaced with matrices of all signatures. We also analyze the distinguished invariant Crofton distribution supported on the minimal orbit, and show that, somewhat surprisingly, it sometimes defines the trivial valuation, thus producing a distribution in the kernel of the cosine transform of particularly small support. In the heart of the paper lies the description by Muro of the | det X| s family of distributions on the space of symmetric matrices, which we use to construct a family of O(p, q)-invariant Crofton distributions. We conjecture there are no others, which we then prove for O(p, 2) with p even. The functorial properties of Crofton distributions, which serve an important tool in our investigation, are studied by T. Wannerer and the author in the Appendix.
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