For measures with polylogarithmic Fourier decay and expanding integer matrix sequences, the number of hits of the orbit on shrinking boxes equals the expected count up to an error of order Ψ(N)^(d/(d+1)) (log Ψ(N))^(2+ε).
Inhomogeneous Diophantine Approximation on $M_0$-sets
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abstract
We prove new quantitative Schmidt-type theorem for Diophantine approximations with restraint denominators on fractals (more precisely, on $M_0$-sets). Our theorems introduce a sharp balance condition between the growth rate of the sequence of denominators and the decay rate of the Fourier transform of a Rajchman measure. Among the other things, this allows applications to sequences of denominators of polynomial growth. In particular, we infer new inhomogeneous Khintchine-J\"arnik type theorems with restraint denominators for a broad family of denominator sequences. Furthermore, our results provide non-trivial lower bounds for Hausdorff dimensions of intersections of two sets of inhomogeneously well-approximable numbers with restraint denominators.
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Quantitative Matrix-Driven Diophantine approximation on $M_0$-sets
For measures with polylogarithmic Fourier decay and expanding integer matrix sequences, the number of hits of the orbit on shrinking boxes equals the expected count up to an error of order Ψ(N)^(d/(d+1)) (log Ψ(N))^(2+ε).