While studying analytic aspects of turbulence of fluids and waves, I came across very interesting questions in Diophantine approximation, a beautiful topic that I am not an expert in.
The question that I would like to ask here concerns approximating irrationals using a restricted set of denominators and prescribing their irrationality measure at the same time.
Let me first set notation and context.
Denote by $\mathbb I = (0,1) \setminus \mathbb Q$ the set of irrationals in the unit interval,
by $|\cdot|$ the Lebesgue measure,
by $\mathcal{H}^\alpha$ the $\alpha$-Hausdorff measure
and by $\text{dim}(\cdot)$ the Hausdorff dimension.
Let $s \geq 2$ and define the Diophantine set
$$ A_s = \{x \in \mathbb I \, : \, |x - \frac{p}{q}| < \frac{1}{q^s} \text{ for infinitely many coprime pairs } (p,q) \in \mathbb N \times \mathbb N \}.$$
Denoting the irrationality measure of $x \in \mathbb I$ by $ \mu(x) = \sup\{\, s \, : x \in A_s \, \}$, define also the set
$$ W_s = \{ \, x \, : \, \mu(x) = s \, \}. $$
The classic results for the metric theory of these sets are:
- Dirichlet approximation theorem: $\quad \mathbb{I}\subset A_2$.
In particular, $\text{dim}(A_2) = 1$ and $|A_2|=1$.
- Jarnik-Besicovitch theorem: $ \quad \text{dim}(A_s) = \frac{2}{s}$, and $\mathcal{H}^{2/s}(A_s) = \infty$ for $s > 2$.
- Güting's theorem: $ \quad \text{dim}(W_s) = \frac{2}{s}$ and $ \mathcal{H}^{2/s}(W_s) = \infty$, for $ s > 2$.
I am interested in restricting the denominators $q$ that can be used to approximate numbers $x \in \mathbb I$, which takes us to the context of the Duffin-Schaeffer theorem. Say we allow only denominators in the set $\mathcal{Q} = \{ \, 3^n : n \in \mathbb N \}$, and work with the set
$$ T_s = \{x \in \mathbb I \, : \, |x - \frac{p}{q}| < \frac{1}{q^s} \text{ for infinitely many coprime pairs } (p,q) \in \mathbb N \times \mathcal{Q} \}. $$
One can prove that $|T_s| = 1$ if $s \leq 1$, and that
$$ \text{dim}(T_s) = \frac{1}{s} \qquad \text{ and } \qquad \mathcal{H}^{1/s}(T_s)=\infty, \qquad \text{ for } s > 1, $$
as follows. By the Duffin-Schaeffer theorem [KM], since $\varphi(3^n) = 2\cdot 3^{n-1}$ and since $\sum_{q \in \mathcal{Q}} \frac{\varphi(q)}{q^s} = \sum_{n=1}^\infty \frac{\varphi(3^n)}{3^{ns}} = \frac{2}{3} \, \sum_{n=1}^\infty \frac{1}{3^{n(s-1)}}$ is infinite if and only if $s \leq 1$, we get
$$ |T_s| = \left\{
\begin{array}{ll}
1, & s \leq 1, \\
0, & s > 1.
\end{array}
\right. $$
Then, the result for Hausdorff measures when $s > 1$ follows using the Mass Transference Principle [BV], the same way that Jarnik-Besicovitch can be recovered from Dirichlet.
However, I need to work with numbers $x \in T_s$ that have irrationality $\mu(x) = s$, that is, with the sets
$$ T_s^* = \{ x \in T_s \, : \, \mu(x) = s \} = T_s \cap W_s = T_s \setminus \bigcup_{\epsilon > 0} A_{s+\epsilon}. $$
Of course, since $T_s^* \subset T_s$, we immediately get $\text{dim}(T^*_s) \leq 1/s$.
**My question:** What is $\text{dim}(T^*_s)$ ?
Some preliminary comments:
- My naive intuition makes me think that $\text{dim}(T^*_s) = 1/s$, but I am not an expert and I do not know if this is actually true. Is it reasonable to expect $\text{dim}(T^*_s) = 1/s$? Or what should we expect?
- This intuition is based in Güting's theorem, which asserts that most $x \in A_s$ have actually irrationality $\mu(x) = s$ (by most I mean that the Hausdorff dimension is preserved). Since the nature of $T^*_s$ with respect to $T_s$ is similar to the nature of $W_s$ with respect to $A_s$, this makes me think that the Hausdorff dimension should be equally preserved when restricting $T_s$ to $T^*_s$.
- However, the analogue of the proof of Güting's theorem does not work for $T^*_s$. The problem is basically that while for Güting's theorem we have $\text{dim}(A_s) = 2/s = \text{dim}\Big( \bigcup_{\epsilon > 0}A_{s+\epsilon} \Big)$, in this case $\text{dim}(T_s) = 1/s < 2/s = \text{dim}\Big( \bigcup_{\epsilon > 0}A_{s+\epsilon} \Big)$. So to define $T_S^*$, we are removing a set that is potentially much bigger than the set we start with.
- A bit more precisely: Güting's theorem follows from Jarnik's as follows: $A_s \setminus \bigcup_{\epsilon > 0} A_{s+\epsilon} \subset W_s$. Since the sets $A_s$ are nested, we have $\mathcal{H}^{2/s}(\bigcup_{\epsilon >0} A_{s+\epsilon}) = \lim_{n \to \infty} \mathcal H^{2/s}(A_{s+1/n}) = 0$, and together with $\mathcal{H}^s(A_s) = \infty$ this implies $\mathcal{H}^{2/s}(W_s) = \infty$, so $\text{dim}(W_s) \geq 2/s$. The reverse inequality follows directly from $W_s \subset A_{s-\epsilon}$ for all $\epsilon > 0$.
However, this argument does not work for $T^*_s$, since even if $\mathcal{H}^{1/s}(T_s) = \infty$, we have $\text{dim}\Big(\bigcup_{\epsilon > 0}A_{s+\epsilon} \Big) = 2/s$ and therefore $\mathcal{H}^{1/s}\Big(\bigcup_{\epsilon > 0}A_{s+\epsilon} \Big) = \infty$ as well.
Any comment or insight will be more than welcome!
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References:
- [BV] V. Beresnevich, S. Velani - A mass transference principle and the Duffin–Schaeffer conjecture for Hausdorff measures. Ann. of Math. (2) 164 (2006), no. 3, 971–992. https://www.jstor.org/stable/20160013
- [KM] D. Koukoulopoulos, J. Maynard - On the Duffin-Schaeffer conjecture. Ann. of Math (2) 192 (2020), no. 1, 251-307. https://annals.math.princeton.edu/2020/192-1/p05