Approximation properties of β-expansions
Approximation properties of β-expansions
复制标题
β-展开式的近似性质
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
Simon Baker
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文献类型：
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作者：
Simon Baker
取一个实数。对于一个函数，定义为这样的集合，即对于无穷多个存在一个满足的序列。在Baker中[-展开的近似性质]。[j] . Acta Arith. 168(2015), 269-287]，作者推测，对于勒贝格几乎每一个，条件暗示是完全勒贝格测度之内。在本文中，我们向证明这个猜想迈出了重要的一步。我们证明了给定一个正实数序列，对于几乎每一个勒贝格，该集合在内都是满勒贝格测度。我们还研究了集合的勒贝格测度为零的情况。应用Beresnevich和Velani在[A]中的传质原理和Hausdorff测度的Duffin-Schaeffer猜想。安。的数学。(2) 164(3)(2006), 971-992]，我们得到了关于的Hausdorff维数和Hausdorff测度的一些结果。
Let be a real number. For a function , define to be the set of such that for infinitely many there exists a sequence satisfying . In Baker [Approximation properties of -expansions. Acta Arith. 168 (2015), 269–287], the author conjectured that for Lebesgue almost every , the condition implies that is of full Lebesgue measure within . In this paper we make a significant step towards proving this conjecture. We prove that given a sequence of positive real numbers satisfying , for Lebesgue almost every , the set is of full Lebesgue measure within . We also study the case where in which the set has Lebesgue measure zero. Applying the mass transference principle developed by Beresnevich and Velani in [A mass transference principle and the Duffin–Schaeffer conjecture for Hausdorff measures. Ann. of Math. (2) 164(3) (2006), 971–992], we obtain some results on the Hausdorff dimension and the Hausdorff measure of .