A FROSTMAN-TYPE LEMMA FOR SETS WITH LARGE INTERSECTIONS, AND AN APPLICATION TO DIOPHANTINE APPROXIMATION

作者：Persson Tomas^*; Reeve Henry W J

来源：Proceedings of the Edinburgh Mathematical Society, 2015, 58(2): 521-542.

DOI：10.1017/S0013091514000066

• diophantine approximation
• Hausdorff dimension
• classes with large intersections

We consider classes G(s)([0, 1]) of subsets of [0, 1], originally introduced by Falconer, that are closed under countable intersections, and such that every set in the class has Hausdorff dimension at least s. We provide a Frostman-type lemma to determine if a limsup set is in such a class. Suppose that E = lim supE(n) subset of [0, 1], and that mu(n) are probability measures with support in E-n. If there exists a constant C such that
integral integral vertical bar x - y vertical bar(-s) d mu(n)(x) d mu(n)(y) < C
for all n, then, under suitable conditions on the limit measure of the sequence (mu(n)), we prove that the set E is in the class G(s)([0, 1]).
As an application we prove that, for alpha > 1 and almost all lambda is an element of (1/2, 1), the set
E-lambda(alpha) = {x is an element of [0, 1] : vertical bar x - s(n vertical bar) < 2(-alpha n) infinitely often},
where s(n) is an element of {(1 - lambda)Sigma(n)(k=0) a(k)lambda(k) and a(k) is an element of {0, 1}}, belongs to the class G(s) for s <= 1/alpha. This improves one of our previously published results.

• 出版日期2015-6