Inhomogeneous Diophantine Approximation on M0-Sets with Restricted Denominators

Andrew D. Pollington, Sanju Velani*, Agamemnon Zafeiropoulos, Evgeniy Zorin

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Let F ⊆ [0, 1] be a set that supports a probability measure μ with the property that |_μ(t)| _ (log |t|) −A for some constant A > 0. Let A = (qn)n∈N be a sequence of natural numbers. If A is lacunary and A > 2, we establish a quantitative inhomogeneous Khintchine-type theorem in which (1) the points of interest are restricted to F and (2) the denominators of the "shifted" rationals are restricted to A. The theorem can be viewed as a natural strengthening of the fact that the sequence (qnxmod1)n∈N is uniformly distributed for μ almost all x ∈ F. Beyond lacunary, our main theorem implies the analogous quantitative result for sequences A for which the prime divisors are restricted to a finite set of k primes and A > 2k. Loosely speaking, for such sequences, our result can be viewed as a quantitative refinement of the fundamental theorem of Davenport, Erdös, and LeVeque (1963) in the theory of uniform distribution.

Original languageEnglish
Pages (from-to)8571-8643
Number of pages73
JournalInternational Mathematics Research Notices
Issue number11
Publication statusPublished - 1 Jun 2022

ASJC Scopus subject areas

  • Mathematics(all)


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