On the minimum of a positive definite quadratic form over non-zero lattice points. Theory and applications

Abstract

Let Σd++ be the set of positive definite matrices with determinant 1 in dimension d⩾2. Identifying any two SLd(Z)-congruent elements in Σd++ gives rise to the space of reduced quadratic forms of determinant one, which in turn can be identified with the locally symmetric space Xd:=SLd(Z)\\SLd(R)/SOd(R). Equip the latter space with its natural probability measure coming from a Haar measure on SLd(R). In 1998, Kleinbock and Margulis established sharp estimates for the probability that an element of Xd takes a value less than a given real number δ>0 over the non-zero lattice points Zd\\{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 Σd++. 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 concise introduction to this topic fitted to our needs, we lay the theoretical foundations of the study of some manifolds frequently appearing in the theory of Signal Processing. This is then applied to the recently introduced Integer-Forcing Receiver Architecture channel whose importance stems from its expected high performance. Here, we give sharp estimates for the probabilistic distribution of the so-called Effective Signal-to-Noise Ratio, which is an essential quantity in the evaluation of the performance of this model.