Dtyjlui

Let $alpha$ be an irrational number.
For every $n$ let $z_n$ be the integer closest to the number $alpha n$. Then we can define
$$A(alpha):= sum_{n=1}^infty |alpha n - z_n|.$$

We can also define somewhat similar quantity
$$B(alpha):= sum_{n=1}^infty left(alpha n - lfloor alpha n rfloor right).$$

$A(alpha)$ and $B(alpha)$ are sums of non-negative numbers, so the values can be real numbers of $+infty$.

EDIT: In fact, the above sums are clearly both equal to $+infty$, since $n alpha$ will be infinitely often arbitrarily close to and integer (and also to a number of the form $k+frac12$, $kinmathbb Z$). See, for example, Multiples of an irrational number forming a dense subset (and several other posts on MSE).

I have missed this obvious fact when posting the question. However the sums $A'(alpha)$ and $B'(alpha)$ described below might still be interesting.

I left here the part about $A(alpha)$ and $B(alpha)$, since some users already posted some comments about these sums. If I edited the first part of the post away, those comments would not make sense.

We can also make modifications where we replace the sequence with monotone sequences:
$$
begin{align*}
A'(alpha):=& sum_{n=1}^infty min_{kle n}|alpha k - z_k|;\
B'(alpha):=& sum_{n=1}^infty min_{kle n} left(alpha k - lfloor alpha k rfloorright).\
end{align*}
$$

In some sense, these quantities tell us how far $alpha$ is from rational numbers. (The monotone versions seem more closely related to question whether $alpha$ is rational or irrational, since for rational $alpha$ all but finitely many terms are zero, so the sum must be finite.)

My question is:

• Were the numbers $A'(alpha)$, $B'(alpha)$ studied somewhere?


• Can they be calculated for some specific irrational numbers? For example, can we calculate them for $alpha=ln 2$, $alpha=sqrt2$, $alpha=e$, $alpha=pi$ or some other well-know irrational numbers?


• Do we get $A'(alpha)=infty$ or $B'(alpha)=infty$ for some numbers? If yes, can such numbers be characterized?


• Are they related to irrationality measure or some other ways to measure how irrational a given number $alpha$ is?

To explain what lead me to these sums: I was thinking about question whether there is a counterexample to Minkowski's theorem for unbounded sets. (Such question was asked here. In the meantime I learned here that such example cannot be found.)
So I was trying to take a line $y=alpha x$ which does not contain any lattice points. I tried to replace the segments of this line by wider rectangles - this lead me to try to compute how wide rectangles I can add without hitting some lattice point. This is related to the question how far are the points $(n,nalpha)$ from lattice points.

$newcommand{bivec}[2]{begin{bmatrix} #1 \ #2 end{bmatrix}}$
Excavating. Still the series for $A'(alpha)$ and $B'(alpha)$ both diverge always.

Let $alpha=[a_0;a_1,a_2,ldots]$ be the continued fraction expansion of $alpha$, and $p_n/q_n$ be its $n$-th convergent:
$$bivec{p_{-1}}{q_{-1}}=bivec{1}{0}, bivec{p_0}{q_0}=bivec{a_0}{1}, bivec{p_n}{q_n}=a_nbivec{p_{n-1}}{q_{n-1}}+bivec{p_{n-2}}{q_{n-2}}.$$
Then for $rgeqslant 0$ and $q_rleqslant n<q_{r+1}$ we have $displaystylemin_{kleqslant n}|alpha k-z_k|=|alpha q_r-p_r|$, which gives
$$A'(alpha)=sum_{n=0}^{infty}(q_{n+1}-q_n)cdot|alpha q_n-p_n|>sum_{n=0}^{infty}frac{q_{n+1}-q_n}{q_{n+1}+q_n}$$
by "Theorem 5" here. But $r_n=q_{n+1}/q_n$ can't converge to $1$ because $r_n=a_{n+1}+1/r_{n-1}$.

(Therefore $dfrac{q_{n+1}-q_n}{q_{n+1}+q_n}=dfrac{r_n-1}{r_n+1}$ can't converge to $0$.)

Thus $A'(alpha)=+infty$ for any (irrational) $alpha$, and therefore $B'(alpha)=+infty$ too.