A note on the Hausdorff dimension of some liminf sets appearing in simultaneous Diophantine approximation

Let Q be an infinite set of positive integers. Denote by W_{\tau, n}(Q) (resp. W_{\tau, n}) the set of points in dimension n simultaneously \tau--approximable by infinitely many rationals with denominators in Q (resp. in N*). A non--trivial lower bound for the Hausdorff dimension of the liminf set W_{\tau, n}\W_{\tau, n}(Q) is established when n>1 and \tau >1+1/(n-1) in the case where the set Q satisfies some divisibility properties. The computation of the actual value of this Hausdorff dimension as well as the one--dimensional analogue of the problem are also discussed

Vertical shift and simultaneous Diophantine approximation on polynomial curves

The Hausdorff dimension of the set of simultaneously tau well approximable points lying on a curve defined by a polynomial P(X)+alpha, where P(X) is a polynomial with integer coefficients and alpha is in R, is studied when tau is larger than the degree of P(X). This provides the first results related to the computation of the Hausdorff dimension of the set of well approximable points lying on a curve which is not defined by a polynomial with integer coefficients. The proofs of the results also include the study of problems in Diophantine approximation in the case where the numerators and the denominators of the rational approximations are related by some congruential constraint.Comment: 22

How far can you see in a forest?

We address a visibility problem posed by Solomon & Weiss. More precisely, in any dimension $n := d + 1 \ge 2$, we construct a forest \F with finite density satisfying the following condition : if \e > 0 denotes the radius common to all the trees in \F, then the visibility \V therein satisfies the estimate \V(\e) = O(\e^{-2d-\eta}) for any $\eta > 0$, no matter where we stand and what direction we look in. The proof involves Fourier analysis and sharp estimates of exponential sums.Comment: This is an extended version of a paper to appear. Minor typos have been correcte

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

A Contribution to Metric Diophantine Approximation : the Lebesgue and Hausdorff Theories

This thesis is concerned with the theory of Diophantine approximation from the point of view of measure theory. After the prolegomena which conclude with a number of conjectures set to understand better the distribution of rational points on algebraic planar curves, Chapter 1 provides an extension of the celebrated Theorem of Duffin and Schaeffer. This enables one to set a generalized version of the Duffin–Schaeffer conjecture. Chapter 2 deals with the topic of simultaneous approximation on manifolds, more precisely on polynomial curves. The aim is to develop a theory of approximation in the so far unstudied case when such curves are not defined by integer polynomials. A new concept of so–called “liminf sets” is then introduced in Chapters 3 and 4 in the framework of simultaneous approximation of independent quantities. In short, in this type of problem, one prescribes the set of integers which the denominators of all the possible rational approximants of a given vector have to belong to. Finally, a reasonably complete theory of the approximation of an irrational by rational fractions whose numerators and denominators lie in prescribed arithmetic progressions is developed in chapter 5. This provides the first example of a Khintchine type result in the context of so–called uniform approximation

Liminf Sets in Simultaneous Diophantine Approximation

Let Q be an infinite set of positive integers. Denote by W(Q) the set of n-tuples of real numbers simultaneously tau-well approximable by infinitely many rationals with denominators in Q but only by finitely many rationals with denominators in the complement of Q. The Hausdorff dimension of the liminf set W(Q) is computed when tau > 2 + 1/n. A p-adic analogue of the problem is also studied