Math Problems Vault

Problem 2.166

Let $\theta$ be an irrational number. The numbers

x_{n} = \theta n - \left[\theta n\right]

are equidistributed on the interval $\left[0,1\right]$ .

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Problem 2.167

Let $\theta$ denote an irrational number. Put $\varepsilon_{n}=1$ or $\varepsilon_{n}=0$ according as the integer next to $n\theta$ is larger or smaller than $n\theta$ . Let $a$ and $d$ , $a \geq 0$ , $d > 0$ , stand for two integers. Then we find that

\lim _{n \to \infty } \frac{\sum _{k=0}^{n-1} \varepsilon _{a+kd}}{n} = \frac{1}{2} .

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Problem 2.168

Let $\theta$ be an irrational number and $\alpha$ be defined as $\alpha = q\theta$ , $q$ integer, $q \neq 0$ . As $z$ converges to $e^{2\pi i \alpha }$ along the ray $\arg z = 2\pi \alpha$ , the function

f(z) = \sum _{n=1}^{\infty } (n\theta - \left[n\theta \right]) z^{n}, \qquad z \text{ arbitrary complex}, \ \left|z\right| < 1,

increases to $\infty$ in such a way that

\lim _{r \to 1-0} (1-r) f(re^{2\pi i \alpha }) = \frac{1}{2\pi i q}

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Remark.

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Problem 88 of Part I.

Problem 2.169

Determine for real $x$ the function

f(x) = \lim _{n \to \infty } \frac{\cos ^{2}\pi x + \cos ^{4}2\pi x + \cos ^{6}3\pi x + \cdots + \cos ^{2n} n\pi x}{n} .

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Problem 2.170

The decimal fraction

\theta = 0.123456789101112\underline{13}\ldots

(the natural numbers listed consecutively) represents an irrational number. According to Problem 166 the numbers

n\theta - \left[n\theta \right], \qquad n = 1, 2, 3, \ldots

are everywhere dense on the interval $\left[0,1\right]$ . Show that this is already the case for the subset

10^{n}\theta - \left[10^{n}\theta \right], \qquad n = 0, 1, 2, 3, \ldots

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Problem 2.171

The number

e = 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \cdots + \frac{1}{n!} + \cdots

is irrational (see Problem 258 of Part VIII). Prove that the only limit point of the set

n! \, e - \left[n! \, e\right], \qquad n = 1, 2, 3, \ldots

is zero.

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Problem 2.172

Suppose that the polynomial $P(x) = a_{1}x + a_{2}x^{2} + \cdots + a_{r}x^{r}$ has at least one irrational coefficient. Then the numbers

P(n) - \left[P(n)\right], \qquad n = 1, 2, 3, \ldots

have infinitely many limit points.

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Problem 2.173

Let $\theta$ be an irrational number, $x_{n} = n\theta - \left[n\theta \right]$ , $n = 1, 2, 3, \ldots$ and let $\alpha_{1}, \alpha_{2}, \alpha_{3}, \ldots , \alpha_{n}, \ldots$ be a monotone decreasing sequence of positive numbers whose sum diverges. Then we find for any properly integrable function $f(x)$ on $\left[0,1\right]$ that

\lim _{n \to \infty } \frac{\alpha _{1}f(x_{1}) + \alpha _{2}f(x_{2}) + \cdots + \alpha _{n}f(x_{n})}{\alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}} = \int _{0}^{1} f(x) \, dx .

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