Robert | Searcharxiv

arXiv2018

Sidon sets are proportionally Sidon with small Sidon constants

In his seminal work on Sidon sets, Pisier found an important characterization of Sidonicity: A set is Sidon if and only if it is proportionally quasi-independent. Later, it was shown that Sidon sets were proportionally `special' Sidon in several other ways. Here, we prove that Sidon sets in torsion-free groups are proportionally $n$-degree independent, a higher order of independence than quasi-independence, and we use this to prove that Sidon sets are proportionally Sidon with Sidon constants arbitrarily close to one, the minimum possible value.

math.FA↗

arXiv2018

Existence of Large Independent-Like Sets

Let $G$ be a compact abelian group and $\Gamma$ be its discrete dual group. For $N \in \mathbb{N}$, we define a class of independent-like sets, $N$-PR sets, as a set in $\Gamma$ such that every $\mathbb{Z}_N$-valued function defined on the set can be interpolated by a character in $G$. These sets are examples of $\varepsilon$-Kronecker sets and Sidon sets. In this paper we study various properties of $N$-PR sets. We give a characterization of $N$-PR sets, describe their structures and prove the existence of large $N$-PR sets.

math.FA↗

arXiv2014

On the Capacity of the Two-Hop Half-Duplex Relay Channel

Although extensively investigated, the capacity of the two-hop half-duplex (HD) relay channel is not fully understood. In particular, a capacity expression which can be easily evaluated is not available and an explicit coding scheme which achieves the capacity is not known either. In this paper, we derive a new expression for the capacity of the two-hop HD relay channel by simplifying previously derived converse expressions. Compared to previous results, the new capacity expression can be easily evaluated. Moreover, we propose an explicit coding scheme which achieves the capacity. To achieve the capacity, the relay does not only send information to the destination by transmitting information-carrying symbols but also with the zero symbols resulting from the relay's silence during reception. As examples, we compute the capacities of the two-hop HD relay channel for the cases when the source-relay and relay-destination links are both binary-symmetric channels (BSCs) and additive white Gaussian noise (AWGN) channels, respectively, and numerically compare the capacities with the rates achieved by conventional relaying where the relay receives and transmits in a codeword-by-codeword fashion and switches between reception and transmission in a strictly alternating manner. Our numerical results show that the capacities of the two-hop HD relay channel for BSC and AWGN links are significantly larger than the rates achieved with conventional relaying.

cs.IT↗