On fractal measures and diophantine approximation

Abstract

Abstract. We study diophantine properties of a typical point with respect to measures on R n . Namely, we identify geometric conditions on a measure µ on R n guaranteeing that µ-almost every y ∈ R n is not very well multiplicatively approximable by rationals. Measures satisfying our conditions are called 'friendly'. Examples include smooth measures on nondegenerate manifolds, thus the present paper generalizes the main result of [KM]. Another class of examples is given by measures supported on self-similar sets satisfying the open set condition, as well as their products and pushforwards by certain smooth maps.

Extremality and dynamically defined measures, part I: Diophantine properties of quasi-decaying measures

“…In relation to extremality, we shall prove that every weakly quasi-decaying measure is strongly extremal (Corollary 1.8), thus generalizing the main result of [12] and in particular providing a third proof of Sprindžuk's conjecture. This implication also proves a conjecture of KLW [12, §10.5] that nonplanar and decaying measures are strongly extremal, i.e.…”

“…The definitions given below are easily seen to be equivalent to KLW's original definitions in [12]. Definition 1.1 Let μ be a measure on an open set U ⊆ R d , and let Supp(μ) denote the topological support of μ.…”

“…that the Federer condition is unnecessary in their main theorem. Although the proof of this result uses essentially the full machinery of the existing proofs of Sprindžuk's conjecture [12,14], it is worth noting that the following result (which does not imply Sprindžuk's conjecture) can be proven using only elementary real analysis together with the Simplex Lemma: The idea of proving the extremality of measures using the Simplex Lemma is due to Pollington and Velani [19,Theorem 1]. Proving Theorem 1.4 was a key step in our construction of the definition of the quasi-decay condition, since it allowed us to see what the minimal hypotheses on the measure were such that the proof would work.…”

“…Proving Theorem 1.4 was a key step in our construction of the definition of the quasi-decay condition, since it allowed us to see what the minimal hypotheses on the measure were such that the proof would work. It was only later that we realized the Sprindžuk conjecture machinery developed in [12,14] would work for our measures as well.…”

“…Fix d ∈ N. The quality of rational approximations to a vector x ∈ R d can be measured by its exponent of irrationality, which is defined by the formula ω(x) = lim sup [14], and later strengthened by Kleinbock, Lindenstrauss, and Weiss (hereafter abbreviated "KLW") in [12], who considered a class of measures which they called "friendly" and showed that these measures are strongly extremal. However, their definition is somewhat rigid and many interesting measures, in particular ones coming from dynamics, do not satisfy their condition.…”