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).
Quantitative Diophantine approximation and Fourier dimension of sets: Dirichlet non-improvable numbers versus well-approximable numbers
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abstract
Let $E\subset [0,1]$ be a set that supports a probability measure $\mu$ with the property that $|\widehat{\mu}(t)|\ll (\log |t|)^{-A}$ for some constant $A>2.$ Let $\mathcal{A}=(q_n)_{n\in \N}$ be a positive, real-valued, lacunary sequence. We present a quantitative inhomogeneous Khintchine-type theorem in which the points of interest are restricted to $E$ and the denominators of the shifted fractions are restricted to $\mathcal{A}.$ Our result improves and extends a previous result in this direction obtained by Pollington-Velani-Zafeiropoulos-Zorin (2022). We also show that the Dirichlet non-improvable set VS well-approximable set is of positive Fourier dimension.
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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).