Let
$\unicode[STIX]{x1D6FD}\in (1,2)$
be a real number. For a function
$\unicode[STIX]{x1D6F9}:\mathbb{N}\rightarrow \mathbb{R}_{\geq 0}$
, define
$W_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D6F9})$
to be the set of
$x\in \mathbb{R}$
such that for infinitely many
$n\in \mathbb{N},$
there exists a sequence
$(\unicode[STIX]{x1D716}_{i})_{i=1}^{n}\in \{0,1\}^{n}$
satisfying
$0\leq x-\sum _{i=1}^{n}(\unicode[STIX]{x1D716}_{i}/\unicode[STIX]{x1D6FD}^{i})\leq \unicode[STIX]{x1D6F9}(n)$
. In Baker [Approximation properties of
$\unicode[STIX]{x1D6FD}$
-expansions. Acta Arith. 168 (2015), 269–287], the author conjectured that for Lebesgue almost every
$\unicode[STIX]{x1D6FD}\in (1,2)$
, the condition
$\sum _{n=1}^{\infty }2^{n}\unicode[STIX]{x1D6F9}(n)=\infty$
implies that
$W_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D6F9})$
is of full Lebesgue measure within
$[0,1/(\unicode[STIX]{x1D6FD}-1)]$
. In this paper we make a significant step towards proving this conjecture. We prove that given a sequence of positive real numbers
$(\unicode[STIX]{x1D714}_{n})_{n=1}^{\infty }$
satisfying
$\lim _{n\rightarrow \infty }\unicode[STIX]{x1D714}_{n}=\infty$
, for Lebesgue almost every
$\unicode[STIX]{x1D6FD}\in (1.497,\ldots ,2)$
, the set
$W_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D714}_{n}\cdot 2^{-n})$
is of full Lebesgue measure within
$[0,1/(\unicode[STIX]{x1D6FD}-1)]$
. We also study the case where
$\sum _{n=1}^{\infty }2^{n}\unicode[STIX]{x1D6F9}(n)<\infty$
in which the set
$W_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D6F9})$
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
$W_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D6F9})$
.