Abstract
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 .
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
