COUNTEREXAMPLES TO MINKOWSKI'S CONJECTURE AND ESCAPE OF MASS IN POSITIVE CHARACTERISTIC (original) (raw)

Counterexamples to Minkowski's Uniqueness Conjecture and Escape of Mass in Positive Characteristic

arXiv (Cornell University), 2023

We show that there are infinitely many counterexamples to Minkowski's conjecture in positive characteristic regarding uniqueness of the upper bound of the multiplicative covering radius, µ, by constructing a sequence of compact A orbits where µ obtains its conjectured upper bound. In addition, we show that these orbits, as well as a slightly larger sequence of orbits, must exhibit complete escape of mass.

Geometry of Numbers in Positive Characteristic

arXiv (Cornell University), 2023

In this paper, we shall discuss topics in geometry of numbers in the positive characteristic setting, such as covering radii. We find a closed form for covering radii with respect to convex bodies, which will lead to a proof of the positive characteristic analogue of Woods' conjecture in this setting. Then, we will prove a positive characteristic analogue of Minkowski's conjecture about the multiplicative covering radius. To do this, we shall prove a positive characteristic analogue of Solan's result that every diagonal orbit intersects the set of well rounded lattices. This implies that the Gruber-Mordell spectrum in positive characteristic is trivial in every dimension.

A note on a conjecture of Poincar�

Celestial Mechanics, 1981

We prove the following weakened version of Poincar6's conjecture on the density of periodic orbits of the restricted three-body problem : The measure of Lebesgue of the set of bounded orbits which are not contained in the closure of the set of periodic orbits goes to zero when the mass parameter does.

A note on Eckmann-Ruelle's conjecture

arXiv: Dynamical Systems, 2015

We introduce a class of C1+alphaC^{1+\\alpha}C1+alpha isolated nonuniformly hyperbolic sets Lambda\\LambdaLambda for which supmuinmathcalMfh(mu)−chi+(mu)\\sup_{\\mu \\in {\\mathcal M}_f}\\{h(\\mu) - \\chi^+(\\mu)\\}supmuinmathcalMfh(mu)−chi+(mu) equals the rate of escape from Lambda\\LambdaLambda, where chi+(mu)\\chi^+(\\mu)chi+(mu) is the average of the sum of positive Lyapunov exponents counted with their multiplicity.

On perfect weighted coverings with small radius

1992

We extend the results of our previous paper [8] to the nonlinear case: The Lloyd polynomial of the covering has at least R distinct roots among 1,..., n, where R is the covering radius. We investigate PWC with diameter 1, finding a partial characterization. We complete an investigation begun in [8] on linear PMC with distance 1 and diameter 2.

On packing of Minkowski balls. II

arXiv (Cornell University), 2023

This is the continuation of the author's ArXiv presentation ''On packing of Minkowski balls. I". In section 2 we investigate lattice packings of Minkowski balls and domains. By results of the proof of Minkowski conjecture about the critical determinant we devide the balls and domains on 3 classes: Minkowski, Davis and Chebyshev-Cohn. The optimal lattice packings of the balls and domains are obtained. The minimum areas of hexagons inscribed in the balls and domains and circumscribed around their are given. Direct limits of direct systems of Minkowski balls and domains and their critical lattices are calculated.

The geometry of Minkowski spaces — A survey. Part I

Expositiones Mathematicae, 2001

We survey elementary results in Minkowski spaces (i.e. finite dimensional Banach spaces) that deserve to be collected together, and give simple proofs for some of them. We place special emphasis on planar results. Many of these results have often been rediscovered as lemmas to other results. In Part I we cover the following topics: The triangle inequality and consequences such as the monotonicity lemma, geometric characterizations of strict convexity, normality (Birkhoff orthogonality), conjugate diameters and Radon curves, equilateral triangles and the affine regular hexagon construction, equilateral sets, circles: intersection, circumscribed, characterizations, circumference and area, inscribed equilateral polygons.

On invariant measures, minimal sets and a lemma of margulis

Inventiones Mathematicae, 1979

In [3] the author obtained a classification of all finite invariant measures of a certain class of horospherical flows. In the present paper we show that if F is an 'arithmetic' lattice then every locally finite ergodic invariant measure of the action of any unipotent subgroup (a horospherical subgroup as above is always unipotent) is necessarily finite. The first step is the following theorem. (0.1) Theorem. Let {ut}t~ ~ be a one-parameter group of unipotent matrices in SL(n, IR). 7hen every locally finite, ergodic, {u~}-invariant measure on SL(n, IR)/SL(n, 2g) is finite. Theorem 0.1 is closely related to the following result in [7] generally known as ' Margulis's lemma'. (0.2) Theorem. Let {ut}t~be as in Theorem 0.1. Then for any x~SL(n, IR)/SL(n, Z) the 'positive semi-orbit' {u,x[ t_->O} does not tend to infinity. That is, there exists a compact subset K of SL(n, 1R)/SL(n, 7l) such that {t >= O lUtXEK } is unbounded. Certainly, in view of Theorem 0.1 for any xeSL(n, IR)/SL(n,~.) the positive semi-orbit and the negative semi-orbit cannot both tend to infinity. For otherwise the 'time' measure along the orbit would be an ergodic, locally finite measure, which is invariant under the flow but not finite. On the other hand our proof of Theorem 0.1 involves finding a compact set K, for the given x, such that the set {t > O lu, xeK} has positive density (cf. Theorem * Supported in part by a National Science Foundation grant (USA)