Quantitative Diophantine approximation and Fourier dimension of sets: Dirichlet non-improvable numbers versus well-approximable numbers

Let $E\subset [0,1]$ be a set that supports a probability measure $\mu$ with the property that $\mathrm{\mid }\widehat{\mu }(t)\mathrm{\mid }\ll (\log \mathrm{\mid }t\mathrm{\mid }{)}^{-A}$ for some constant $A>2.$ Let $\mathcal{A}=(q_n{)}_{n\in \mathbb{N}}$ 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 $\mathcal{A}\mathrm{.}$ 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.