Quantitative Diophantine approximation and Fourier dimension of sets: Dirichlet non-improvable numbers versus well-approximable numbers
Abstract
Let E⊂ [0,1] be a set that supports a probability measure μ with the property that |μ(t)| ( |t|)-A for some constant A>2. Let A=(qn)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 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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