We study the distribution of sequences of the form (qnβy)n=1ββ,
where (qnβ)n=1ββ is some increasing sequence of integers. In
particular, we study the Lebesgue measure and find bounds on the Hausdorff
dimension of the set of points Ξ³β[0,1) which are well approximated
by points in the sequence (qnβy)n=1ββ. The bounds on Hausdorff
dimension are valid for almost every y in the support of a measure of
positive Fourier dimension. When the required rate of approximation is very
good or if our sequence is sufficiently rapidly growing, our dimension bounds
are sharp. If the measure of positive Fourier dimension is itself Lebesgue
measure, our measure bounds are also sharp for a very large class of sequences.
We also give an application to inhomogeneous Littlewood type problems.Comment: 15 pages. Niclas Technau pointed out to us that Theorems 1 and 2 in
the original version were in fact consequences of arXiv:2307.14871 . In this
revised version, we have strengthened both theorems to cover a wider class of
sequence