Can I read On the Minimum of a Positive Definite Quadratic Form over Non--Zero Lattice points. Theory and Applications on EtoBox?

On the Minimum of a Positive Definite Quadratic Form over Non--Zero Lattice points. Theory and Applications by Adiceam, Faustin; Zorin, Evgeniy is a scholarly article available to read on EtoBox.

What is On the Minimum of a Positive Definite Quadratic Form over Non--Zero Lattice points. Theory and Applications about?

Let $\Sigma_d^{++}$ be the set of positive definite matrices with determinant 1 in dimension $d\ge 2$. Identifying any two $SL_d(\mathbb{Z})$-congruent elements in $\Sigma_d^{++}$ gives rise to the space of reduced quadratic forms of determinant one, which in turn can be identified with the locally symmetric space $X_d:=SL_d(\mathbb{Z})\backslash SL_d(\mathbb{R})/SO_d(\mathbb{R})$. Equip the latter space with its natural probability measure coming from a Haar measure on $SL_d(\mathbb{R})$. In 1998, Kleinbock and Margulis established sharp estimates for the probability that an element of $X_d$ takes a value less than a given real number $\delta>0$ over the non--zero lattice points $\mathbb{Z}^d\backslash\{ 0 \}$. In this article, these estimates are extended to a large class of probability measures arising either from the spectral or the Cholesky decomposition of an element of $\Sigma_d^{++}$. The sharpness of the bounds thus obtained are also established (up to multiplicative constants) for a subclass of these measures. Although of an independent interest, this theory is partly developed here with a view towards application to Information Theory. More precisely, after providing a c

Author

Adiceam, Faustin; Zorin, Evgeniy

Published

2016

Language

EN