The quantitative version of Kchintchin's theorem proved by W.Schmidt states that for a.e. $x\in[0,1]$ and every positive integer $K$, if we denote the number of solutions $(p,q)$ to the inequality
$$|x-\frac{p}{q}|<\frac{\psi(q)}{q}$$ with $0<q<K$ and $0<p<q$, by $N(x,K)$ then $$N(x,K)=\Psi(K)+C(x)\Psi^{\frac{1}{2}}(K)\Omega^{\frac{1}{2}}(K)\log^{3}(\Psi(K))$$ where $\Psi(K)=\sum_{n=1}^{K}\psi(n)$ and $\Omega(K)=\sum_{n=1}^{K}\frac{\psi(n)}{n}$.
Of course the result holds under the assumption that $\psi(n)$ is monotonically non-increasing.
I wish to understand what is the measure of the set for which the main term in the expression above is the dominant one (for large enough $K$), i.e. $$|\{x\in[0,1]|\,N(x,K)\leq c\cdot\Psi(K)\}|$$ where $c$ is some absolute constant independent of $x$ and $K$.
For my purpose it is enough to consider $\psi(n)=\frac{1}{n^\alpha}$ with some $\alpha<1$ (in fact I'm interested in the case $\alpha\in(0,\frac{1}{3})$).
Also, if it makes more sense or easier to estimate I would be happy with an upper bound on the measure of the set $$|\{x\in[0,1]|\,C(x)\leq c\cdot K^{\frac{1}{2}}\}|$$

It is clear that the measure of both sets tends to one as $K\rightarrow \infty$. So I'm really interested in the rate of convergence and its dependence on $\alpha$ (or, in the general case, in $\psi$).
Another way to think about this problem is that I wish to replace $C(x)$ by a uniform constant, which might only hold for a all $x$ outside some exceptional set. I'm trying to estimate the size of this exceptional set.

  • $\begingroup$ Erm... The measure of the set $|\{x\in [0,1]: C(x)\le cK^{\frac12}\}|$ for what $K$? $\endgroup$ – fedja Jun 3 '18 at 23:35
  • $\begingroup$ For all sufficiently large $K\in\mathbb{N}$. $\endgroup$ – Itay Jun 4 '18 at 4:33
  • $\begingroup$ Any number $x$ for which $C(x)<+\infty$ satisfies the predicate "$C(x)<cK^{1/2}$ for all sufficiently large $K\in\mathbb N$". So, what set are you interested in? $\endgroup$ – fedja Jun 4 '18 at 4:48
  • $\begingroup$ As I added in my edit, I'm interested in the rate at which the measure set $$|\{x \in [0,1] |C(x) \geq cK^{\frac{1}{2}}\}|$$ decays to zero $\endgroup$ – Itay Jun 4 '18 at 4:55

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