Related papers: Number Theory meets Wireless Communications: an in…
Electromagnetic information theory (EIT) is one of the emerging topics for 6G communication due to its potential to reveal the performance limit of wireless communication systems. For EIT, one of the most important research directions is…
This work is motivated by problems on simultaneous Diophantine approximation on manifolds, namely, establishing Khintchine and Jarnik type theorems for submanifolds of R^n. These problems have attracted a lot of interest since Kleinbock and…
In this paper we consider the probabilistic theory of Diophantine approximation in projective space over a completion of Q. Using the projective metric studied by Bombieri, van der Poorten, and Vaaler we prove the analogue of Khintchine's…
As wireless technology begins to utilize physically larger arrays and/or higher frequencies, the transmitter and receiver will reside in each other's radiative near field. This fact gives rise to unusual propagation phenomena such as…
In this paper we develop a general theory of metric Diophantine approximation for systems of linear forms. A new notion of `weak non-planarity' of manifolds and more generally measures on the space of $m\times n$ matrices over $\Bbb R$ is…
Degrees of freedom (DoF) gains are studied in wireless networks with cooperative transmission under a backhaul load constraint that limits the average number of messages that can be delivered from a centralized controller to base station…
Multiple-antenna broadcast channels with $M$ transmit antennas and $K$ single-antenna receivers is considered, where the channel of receiver $r$ takes one of the $J_r$ finite values. It is assumed that the channel states of each receiver…
In this paper, we show that the total Degrees-Of-Freedoms (DOF) of the $K$-user Gaussian Interference Channel (GIC) can be achieved by incorporating a new alignment technique known as \emph{real interference alignment}. This technique…
In this paper, we study inhomogeneous Diophantine approximation with rational numbers of reduced form. The central object to study is the set $W(f,\theta)$ as follows, \begin{eqnarray*} \left\{x\in [0,1]:\left…
In this paper the metric theory of Diophantine approximation associated with the small linear forms is investigated. Khintchine-Groshev theorems are established along with Hausdorff measure generalization without the monotonic assumption on…
In 1996 N. Chevallier proved a beautiful lemma which connects Diophantine approximation and multidimensional generalizations of the famous Three Distance Theorem. Using this lemma we show how known results about multidimensional three…
We investigate the number of integer solutions to a multiplicative Diophantine approximation problem and show that the associated counting function converges in distribution to a normal law. Our approach relies on the analysis of…
In these notes, we begin by recalling aspects of the classical theory of metric Diophantine approximation; such as theorems of Khintchine, Jarn\'{\i}k, Duffin-Schaeffer and Gallagher. We then describe recent strengthening of various…
Let $E\subset [0,1]$ be a set that supports a probability measure $\mu$ with the property that $|\widehat{\mu}(t)|\ll (\log |t|)^{-A}$ for some constant $A>2.$ Let $\mathcal{A}=(q_n)_{n\in \N}$ be a positive, real-valued, lacunary sequence.…
Duffin and Schaeffer have generalized the classical theorem of Khintchine in metric Diophantine approximation in the case of any error function under the assumption that all the rational approximants are irreducible. This result is extended…
Diophantine exponents are ones of the simplest quantitative characteristics responsible for the approximation properties of linear subspaces of a Euclidean space. This survey is aimed at describing the current state of the area of…
The metrical theory of the product of consecutive partial quotients is associated with the uniform Diophantine approximation, specifically to the improvements to Dirichlet's theorem. Achieving some variant forms of metrical theory in…
The degrees of freedom (DoF) number of the fully connected K-user Gaussian interference channel is known to be K/2. In [1], the DoF for the same channel model was studied while allowing each message to be available at its own transmitter as…
Many results related to quantitative problems in the metric theory of Diophantine approximation are asymptotic, such as the number of rational solutions to certain inequalities grows with the same rate almost everywhere modulo an asymptotic…
We consider a cooperative X-channel with $\sf K$ transmitters (TXs) and $\sf K$ receivers (Rxs) where Txs and Rxs are gathered into groups of size $\sf r$ respectively. Txs belonging to the same group cooperate to jointly transmit a message…