I am trying to prove the following claim
For given $\varepsilon>0$, we can always find an integer $\tau$ such that $\tau\sqrt{2}$ differs from another integer by less than $\varepsilon$ and furthermore it can be proved that there exist infinitely many such numbers $\tau$, and that the difference between two consecutive ones is bounded. Find a bound like this.
Approximation is easy. The main point here is I want to control the distance between two consecutive ones.
I was trying to use the Dirichlet approximation theorem but I couldn't control the distance between two consecutive ones. Can anyone help me with this?
Suppose $\tau \sqrt2$ is within $\epsilon$ of an integer, say that it is $n + \delta$ for $n \in \mathbb N$ and $0 < \delta < \epsilon$. (The corresponding case $n - \delta$ is more or less the same.)
Then consider the sequence $$\tau \sqrt2, 2\tau \sqrt2, 3\tau\sqrt2, \dots.$$
Initially, the fractional parts of the numbers in this sequence go up in steps of $\delta$, but after $k \approx \frac1\delta$ steps, you get to a number $k \tau \sqrt2 = kn + k\delta$ whose fractional part $k\delta$ has gotten very close to $1$; one more step and it will overflow. This means that in particular, $k \tau \sqrt2$ is within $\epsilon$ of the integer $kn+1$.
This will continue happening at intervals of approximately $\frac1\delta$. The fractional parts of this sequence increase by $\delta$ at every step, and every time they pass $1$, the corresponding element of the sequence is with $\epsilon$ of an integer.
(Since the multiples of $\tau \sqrt2$ go up in steps of approximately $\frac1\delta$, the multiples of $\sqrt2$ go up in steps of approximately $\frac{\tau}{\delta}$. We are probably skipping lots of good approximations of $\sqrt2$, of course, but this is good enough for your purposes.)
An example. For $\epsilon = 0.1$, we might start with $5 \sqrt2 \approx 7.071$, so that $\delta = 5\sqrt2 - 7 \approx 0.071$.
Then the sequence $$5 \sqrt2, 10\sqrt2, 15\sqrt2, 20\sqrt2, \dots$$ will be approximately $$7.071, 14.142, 21.213, 28.284, \dots$$ with fractional parts going up by about $0.071$.
Eventually these fractional parts get close to $1$. Since $\frac1\delta \approx 14.071$, after $14$ steps we get to $70 \sqrt 2 \approx 98.995$, and after $15$ steps we get to $75 \sqrt2 \approx 106.066$. Both of these are guaranteed to be within $\epsilon$ of an integer, because $|99 - 70\sqrt2| \approx 0.005$ and $|106 - 75\sqrt2| \approx 0.066$ add up to exactly $\delta$.
This will happen every at-most-$15$ multiples of $5\sqrt2$, or every at-most-$75$ multiples of $\sqrt2$.