Diophantine Approximation and Hausdorff Dimension

Ubiquity and large intersections properties under digit frequencies constraints

“…Similar results were obtained in R d for various families {(x n , l n )} n≥1 [8,9,1,23] or in metric spaces enjoying enough self-similar properties to support a monofractal measure [13,14,4], like the middle third Cantor set [26].…”

“…Using l n = l n (m, ), and taking π p = λ and δ p = m+1− m− for every p ≥ 1, we show that m− m+1− ≤ dim A(m + 1, ), which is finer than the lower bound wm+1− m+1− obtained when using l n = l n (w m , ). Nevertheless, the approach in [1] yields dim A(m + 1, ) = 1. All these remarks show that for families of algebraic numbers, our approach does not provide sharp lower bounds unless the following conjecture holds true (see [10] • The family (z + {nα}, 1/n) n≥1,z∈Z .…”

“…Using that I (δ) n ≤ 2 −(j−1)δ when n ∈ T j,k , the previous remarks yield Without loss of generality we restrict our study to [0,1] and suppose that in the definition of the limsup sets we are concerned with, the sequence (x n ) n≥1 is dense in [0, 1] and the sequence (l n ) n≥1 is non-decreasing and converges to 0.…”

On weighted inhomogeneous Diophantine approximation on planar curves

“…The idea and the concept of ubiquity was originally formulated by Dodson, Rynne and Vickers in [13] and coincided in part with the concept of regular systems of Baker and Schmidt ( [3]). Both have proven to be extremely useful in obtaining lower bounds for the Hausdorff dimension of limsup sets.…”

Exceptional Sets in Dynamical Systems and Diophantine Approximation

“…Jarník's original proof and his more general Hausdorff measure result for simultaneous Diophantine approximation relied heavily on arithmetic arguments. Besicovitch's later independent proof was more geometric and was a basis for the widely used regular systems [9] and ubiquitous systems [30]. It follows from the inclusions (11) and the Jarník-Besicovitch theorem that when σ ≥ 0,…”