We obtain new lower bounds on the Hausdorff dimension of distance sets and pinned distance sets of planar Borel sets of dimension slightly larger than 1, improving recent estimates of Keleti and Shmerkin, and of Liu in ...

We obtain box-counting estimates for the pinned distance sets of (dense subsets of) planar discrete Ahlfors-regular sets of exponent s > 1. As a corollary, we improve upon a recent result of Orponen, by showing that if A ...

We prove that if A is a Borel set in the plane of equal Hausdorff and packing dimension s > 1, then the set of pinned distances {|x − y| : y ∈ A} has full Hausdorff dimension for all x outside of a set of Hausdorff dimension ...

We prove new bounds on the dimensions of distance sets and pinned distance sets of planar sets. Among other results, we show that if A ⊂ R2 is a Borel set of Hausdorff dimension s > 1, then its distance set has Hausdorff ...

Epi-Lipschitz sets in normed spaces are represented as sublevel sets of Lipschitz functions satisfying a so-called qualification condition. Canonical representations through the signed distance functions associated with ...

Average distances are widely used in many fields for calculating the distances between two sets of elements. This paper presents several new average distances by using the ordered weighted average, the probability and the ...

Given a pair of planar curves, one can define its generalized area distance, a concept that generalizes the area distance of a single curve. In this paper, we show that the generalized area distance of a pair of planar ...

Aim: The drivers of phylogenetic beta diversity include both local processes (e.g. environmental filtering) and regional processes (e.g. dispersal limitation). The role of environmental filtering can be investigated more ...