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Dirichlet's theorem says that all numbers $x\in [0,1]$ can be approximated by infinitely many fractions $p/q \in \mathbb{Q}$ with error $|x - p/q| \le 1/q^2$.

I am interested in the following question: Fix $m>0$. Let $S(m)$ denote the set of $x\in [0,1]$ for which no fraction $p/q \in \mathbb{Q}$ with $q<100\sqrt{m}$ approximates $x$ with error smaller than $1/m$, i.e., $|x-p/q| > 1/m$ for any such choice of $p,q$. What bounds are there for the measure of $S(m)$?

EDIT: Let $\Phi(q)$ denote the set of numbers smaller than $q$ coprime to $q$. And let $$I(q) = \bigcup_{p\in \Phi(q)} [p/q, p/q+1/m]$$ For $q_1,q_2<\sqrt{m}$ the sets $I(q_1),I(q_2)$ will be disjoint. This can give a simple lower bound (by analyzing $\sum_{k<\sqrt{m}} \phi(k)$, which is a well known problem). I'm interested in if anything stronger than this bound can be given.

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  • $\begingroup$ Note that if $m<40000$ then $S(m)$ is empty since for any $x$ and any $q$ there exists $p$ such that $|x-(p/q)|\le1/(2q)$. $\endgroup$ May 31, 2023 at 13:03
  • $\begingroup$ It is about some constant depending on 100. Small if 100 is large. What kind of bound (and from which side) would make you happy? $\endgroup$ Jun 1, 2023 at 20:27
  • $\begingroup$ I guess I'm really interested in the asymptotic behavior, i.e., if we replace 100 with a parameter. $\endgroup$ Jun 1, 2023 at 20:33
  • $\begingroup$ So, you take large parameter $T$ (for 100), and after that asymptotics in $m$, right? $\endgroup$ Jun 2, 2023 at 6:30
  • $\begingroup$ yeah, that is what I was thinking $\endgroup$ Jun 3, 2023 at 0:29

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Fix $T$ ($T=100$ in your example) and denote by $S_T(n)$ the set of numbers $x\in [0,1]$ such that $|x-p/q|>1/n^2$ whenever $p,q$ are integers and $1\leqslant q\leqslant Tn$ (so, my $n$ is your $\sqrt{m}$).

Let $0=r_0<r_1<\ldots<r_K=1$ be Farey sequence of all irreducible fractions with denominator at most $Tm$. Denote $\delta_i=r_{i+1}-r_i$ for $i=0,1,\ldots,K-1$ the gap between corresponding Farey fractions. Then $S_T(n)\cap [r_{i},r_{i+1}]$ has length $\max(\delta_i-2/n^2,0)$, so, the measure of $S_T(n)$ equals $$ f(T,n):=\sum_{i:\delta_i>2/n^2} (\delta_i-2/n^2). $$ Recall that if $r_i=a/b$, $r_{i+i}=c/d$ are irreducible fractions, then $\delta_i=1/(bd)$ and $b+d>Tn$. Denote $M=\max(b,d)$, $m=\min(b,d)$. Then $M> Tn/2$, and the inequality $1/(Mm)=\delta_i>2/n^2$ yields $1/m>2M/n^2>T/n$, so, $m<n/T$. And $\delta_i=1/(mM)\leqslant 2/(Tnm)$. Thus, for fixed $m$, the fractions with denominator $m$ give a contribution at most $$ 2\varphi(m)\left(\frac{2}{Tnm}-\frac2{n^2}\right)\leqslant \frac4{Tn}. $$ Summing up over $m<n/T$ gives you at most $4/T^2$.

On the other hand, if $m<n/(3T)$, say, then $mM\leqslant mTn\leqslant n^2/3$ and thus $1/bd\geqslant 3/n^2$. Therefore, each of these intervals (with $m<n/(3T)$) give contribution at least $1/n^2$, and there are about $$2\sum_{m<n/(3T)} \varphi(m)\sim \zeta(2)\frac{n^2}{9T^2}$$ such intervals that gives lower bound also of order $1/T^2$.

So, $$c_1/T^2<f(T,n)<c_2/T^2$$ for some universal explicit constants $c_1,c_2$. If you care on sharp constant, please let me know, this requires more accurate analysis but looks doable.

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  • $\begingroup$ Super cool, the $\Theta(1/T^2)$ is perfect. Thanks so much! $\endgroup$ Jun 4, 2023 at 0:16

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