We address a visibility problem posed by Solomon & Weiss [26]. More precisely, in any dimension n := d + 1 ≥ 2, we construct a forest F with finite density satisfying the following condition : if > 0 denotes the radius common to all the trees in F, then the visibility V therein satisfies the estimate V( ) = O η,d −2d−η for any η > 0, no matter where we stand and what direction we look in. The proof involves Fourier analysis and sharp estimates of exponential sums.
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 this paper. Results are both, on the one hand, from a metrical and a non-metrical point of view and, on the other, from an asymptotic and also a uniform point of view. The principal novelty is a Khintchine type theorem for uniform approximation in this context. Some applications of this theory are also discussed.
This paper is motivated by recent applications of Diophantine approximation in electronics, in particular, in the rapidly developing area of Interference Alignment. Some remarkable advances in this area give substantial credit to the fundamental Khintchine-Groshev Theorem and, in particular, to its far reaching generalisation for submanifolds of a Euclidean space. With a view towards the aforementioned applications, here we introduce and prove quantitative explicit generalisations of the Khintchine-Groshev Theorem for nondegenerate submanifolds of R n . The importance of such quantitative statements is explicitly discussed in Jafar's monograph [13, §4.7.1].
Dense forests are discrete subsets of Euclidean space which are uniformly close to all sufficiently long line segments. The degree of density of a dense forest is measured by its visibility function. We show that cut‐and‐project quasicrystals are never dense forests, but their finite unions could be uniformly discrete dense forests. On the other hand, we show that finite unions of lattices typically are dense forests, and give a bound on their visibility function, which is close to optimal. We also construct an explicit finite union of lattices which is a uniformly discrete dense forest with an explicit bound on its visibility.
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