[Paper Review] Inhomogeneous Diophantine Approximation on $M_0$-sets with restricted denominators

Andrew Pollington, Sanju Velani+2 more|arXiv (Cornell University)|Jun 4, 2019

Mathematical Dynamics and Fractals16 references4 citations

TL;DR

This paper establishes a quantitative inhomogeneous Diophantine approximation theorem on $M_0$-sets with restricted denominators, showing that for lacunary sequences $\mathcal{A} = (q_n)$ and measures $\mu$ with Fourier decay $|\widehat{\mu}(t)| \ll (\log|t|)^{-A}$ for $A > 2$, the sequence $(q_n x \mod 1)$ hits shrinking balls around a fixed $\gamma$ the expected number of times for $\mu$-almost all $x$ in the support of $\mu$. The result extends classical uniform distribution theory to shrinking targets and restricted denominators.

ABSTRACT

Let $F \subseteq [0,1]$ be a set that supports a probability measure $μ$ with the property that $ |\widehatμ(t)| \ll (\log |t|)^{-A}$ for some constant $ A > 0 $. Let $\mathcal{A}= (q_n)_{n\in \mathbb{N}} $ be a sequence of natural numbers. If $\mathcal{A}$ is lacunary and $A >2$, we establish a quantitative inhomogeneous Khintchine-type theorem in which (i) the points of interest are restricted to $F$ and (ii) the denominators of the `shifted' rationals are restricted to $\mathcal{A}$. The theorem can be viewed as a natural strengthening of the fact that the sequence $(q_nx { m \ mod \, } 1)_{n\in \mathbb{N}} $ is uniformly distributed for $μ$ almost all $x \in F$. Beyond lacunary, our main theorem implies the analogous quantitative result for sequences $\mathcal{A}$ for which the prime divisors are restricted to a finite set of $k$ primes and $A > 2k$.

Motivation & Objective

• To extend classical inhomogeneous Diophantine approximation to $M_0$-sets with restricted denominators.
• To establish a quantitative version of the Davenport–Erd\'os–LeVeque theorem for lacunary sequences and shrinking targets.
• To determine the decay rate of the Fourier transform of a measure on an $M_0$-set that ensures uniform distribution modulo one for $\mu$-almost all $x$.
• To generalize results beyond lacunary sequences to sequences with prime divisors restricted to a finite set of $k$ primes, under a stronger decay condition $A > 2k$.

Proposed method

• Use of the Davenport–Erd\'os–LeVeque criterion to link uniform distribution of $ (q_n x \mod 1) $ to the decay rate of the Fourier transform $ \widehat{\mu}(t) $.
• Application of the $\mu$-measure of $W^*$-sets to characterize sets where uniform distribution fails.
• Introduction of a shrinking target function $ \psi(q_n) $ to model shrinking balls $ B(\gamma, \psi(q_n)) $ around a fixed $ \gamma \in [0,1] $.
• Use of summation estimates involving $ s_n = q_n - q_{n-1} $ and partial sums $ S_n $ to control the number of hits in shrinking intervals.
• Employment of logarithmic and harmonic-type bounds on $ \sum s_n / S_n $ and $ \sum s_n / S_n^2 $ to derive convergence conditions.
• Derivation of a quantitative estimate for the discrepancy of $ (q_n x \mod 1) $ in relation to the measure $ \mu $ and the target size $ \psi(q_n) $.

Experimental results

Research questions

• RQ1 Under what conditions on the Fourier decay of $ \mu $ does the sequence $ (q_n x \mod 1) $ uniformly distribute modulo one for $ \mu $-almost all $ x \in F $?
• RQ2 Can the classical inhomogeneous Khintchine theorem be strengthened to allow shrinking targets when denominators are restricted to a lacunary sequence $ \mathcal{A} $?
• RQ3 What is the minimal decay rate $ |\widehat{\mu}(t)| \ll (\log|t|)^{-A} $ required for the expected hitting frequency of shrinking balls $ B(\gamma, \psi(q_n)) $?
• RQ4 How does the result extend from lacunary sequences to sequences with prime divisors restricted to a finite set of $ k $ primes?

Key findings

• For a lacunary sequence $ \mathcal{A} $ and $ \mu $ with $ |\widehat{\mu}(t)| \ll (\log|t|)^{-A} $ for $ A > 2 $, the sequence $ (q_n x \mod 1) $ hits the shrinking ball $ B(\gamma, \psi(q_n)) $ the expected number of times for $ \mu $-almost all $ x \in F $.
• The result implies that $ \mu $-almost all $ x \in F $ are normal numbers, provided $ |\widehat{\mu}(t)| \ll (\log\log|t|)^{-(1+\epsilon)} $ for some $ \epsilon > 0 $.
• For sequences $ \mathcal{A} $ with prime divisors restricted to $ k $ primes, the result holds if $ A > 2k $, extending beyond lacunary sequences.
• The key technical tool is a bound on the sum $ \sum_{k=a}^b \frac{s_k}{\tilde{S}_k} $ with $ \tilde{S}_k = \max(\gamma, S_k) $, which controls the discrepancy in shrinking targets.
• The paper establishes a quantitative version of the Davenport–Erd\'os–LeVeque theorem for restricted denominators and shrinking targets, improving on classical results.
• The proof relies on estimating $ \sum s_k / S_k $ and $ \sum s_k / S_k^2 $ using logarithmic and harmonic-type inequalities, leading to convergence of the discrepancy series.