Multiplicity estimate for solutions of extended Ramanujan's system

We establish a new multiplicity lemma for solutions of a differential system extending Ramanujan's classical differential relations. This result can be useful in the study of arithmetic properties of values of Riemann zeta function at odd positive integers (Nesterenko, 2011)

Explicit bounds for rational points near planar curves and metric Diophantine approximation

The primary goal of this paper is to complete the theory of metric Diophantine approximation initially developed in [Ann. of Math.(2) 166 (2007), p.367-426] for $C^3$ non-degenerate planar curves. With this goal in mind, here for the first time we obtain fully explicit bounds for the number of rational points near planar curves. Further, introducing a perturbational approach we bring the smoothness condition imposed on the curves down to $C^1$ (lowest possible). This way we broaden the notion of non-degeneracy in a natural direction and introduce a new topologically complete class of planar curves to the theory of Diophantine approximation. In summary, our findings improve and complete the main theorems of [Ann. of Math.(2) 166 (2007), p.367-426] and extend the celebrated theorem of Kleinbock and Margulis appeared in [Ann. of Math.(2), 148 (1998), p.339-360] in dimension 2 beyond the notion of non-degeneracy.Comment: 24 page

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

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 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 \emph{Effective Signal--to--Noise Ratio}, which is an essential quantity in the evaluation of the performance of this model

Multiplicity Estimates for Algebraically Dependent Analytic Functions

We prove a new general multiplicity estimate applicable to sets of functions without any assumption on algebraic independence. The multiplicity estimates are commonly used in determining measures of algebraic independence of values of functions, for instance within the context of Mahler's method. For this reason, our result provides an important tool for the proofs of algebraic independence of complex numbers. At the same time, these estimates can be considered as a measure of algebraic independence of functions themselves. Hence our result provides, under some conditions, the measure of algebraic independence of elements in ${\bf F}_q[[T]]$, where ${\bf F}_q$ denotes a finite field.Comment: arXiv admin note: substantial text overlap with arXiv:1103.117

Algebraic Independence and Mahler's method

We give some new results on algebraic independence within Mahler's method, including algebraic independence of values at transcendental points. We also give some new measures of algebraic independence for infinite series of numbers. In particular, our results furnishes, for $n\geq 1$ arbitrarily large, new examples of sets (\theta_1,...,\theta_n)\in\mrr^n normal in the sense of definition formulated by Grigory Chudnovsky (1980).Comment: 6 page

On irrationality measure of Thue-Morse constant

We provide a non-trivial measure of irrationality for a class of Mahler numbers defined with infinite products which cover the Thue-Morse constant