Math Problems Vault

Problem I.3.1

Let $x \in \mathbb {R}$ . Prove the equivalence of the statements

$x \notin \mathbb {Q}$ ;
there exist infinitely many irreducible fractions $\frac{p}{q}$ ( $q \in \mathbb {N}, p \in \mathbb {Z}$ ) such that $\left|x-\frac{p}{q}\right|<\frac{1}{q^{2}}$ ;
there exists a sequence of fractions $\frac{p_{k}}{q_{k}}\left(q_{k} \in \mathbb {N}, p_{k} \in \mathbb {Z}\right)$ such that $q_{k} \rightarrow +\infty$ and $0 \neq x-\frac{p_{k}}{q_{k}}=o\left(\frac{1}{q_{k}}\right)$ as $k \rightarrow +\infty$ .

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Problem I.3.2

We use the notation $\delta (x)=x-\left[x\right]$ for the fractional part of a real number $x$ . Let $x$ be an irrational number. Prove that the sequence $\left\{ \delta (n x)\right\}$ is dense in $(0,1)$ .

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Problem I.3.3

(a)(b)

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(a)

Prove that the sequences $\left\{ \sin n^{2}\right\}$ and $\left\{ \sin 4^{n}\right\}$ have no limits;

(b)

prove that if the sequence $\left\{ \sin \left(2^{n} \theta \right)\right\}$ has a limit, then it equals zero and $\theta =\frac{\pi p}{2^{q}}$ , where $q \in \mathbb {N}, p \in \mathbb {Z}$ .

Problem I.3.4

We use the notation $\delta (x)=x-\left[x\right]$ for the fractional part of a real number $x$ . Let $x \in \mathbb {R}$ . Prove the equivalence of the statements

$x \in \mathbb {Q}$ ;
the set $\left\{ \delta \left(n^{2005} x\right) \mid n \in \mathbb {N}\right\}$ is finite.

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Problem I.3.5

We use the notation $\delta (x)=x-\left[x\right]$ for the fractional part of a real number $x$ . Show by examples that the sequence $\left\{ \delta \left(10^{n} x\right)\right\}$ can be everywhere dense, and can be infinite but nowhere dense in $(0,1)$ .

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Problem I.3.6

Find the sets of partial limits of the sequences

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(a)

$\left\{ \sin n^{2 / 3}\right\}$ ;

(b)

$\left\{ \sqrt[4]{n} \sin \sqrt{n}\right\}$ ;

(c)

$\left\{ \sin \left(\pi n^{3 / 2}\right)\right\}$ ;

(d)

$\left\{ \sin \ln n\right\}$ ;

(e)

$\left\{ \sin (\pi n \ln n)\right\}$ .

Problem I.3.7

(a)(b)(c)(d)

We use the notation $\delta (x)=x-\left[x\right]$ for the fractional part of a real number $x$ . Find the sets of partial limits of the sequences

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(a)

$\left\{ \delta (n x)\right\}$ , where $x \in \mathbb {Q}$ ;

(b)

$\left\{ \delta (n x)\right\}$ , where $x \in \mathbb {R} \backslash \mathbb {Q}$ ;

(c)

$\left\{ \delta \left(n^{\alpha }\right)\right\}$ , where $\alpha \in (0,1)$ ;

(d)

$\left\{ \delta \left(n^{5 / 2}\right)\right\}$ .

Problem I.3.8

We use the notation $\delta (x)=x-\left[x\right]$ for the fractional part of a real number $x$ . Does there exist a real number $x$ such that $\delta \left(x^{n}\right) \in \left[\frac{1}{3}, \frac{2}{3}\right]$ for all indices $n$ ?

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Problem I.3.9

We use the notation $\delta (x)=x-\left[x\right]$ for the fractional part of a real number $x$ . Let $f \in C(\left[0,1\right])$ and $x \in \mathbb {R} \backslash \mathbb {Q}$ . Prove that

\frac{1}{n} \sum _{1 \leqslant k \leqslant n} f(\delta (k x)) \rightarrow \int _{0}^{1} f(t) d t \quad \text{ as } \quad n \rightarrow +\infty .

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Problem I.3.10

Let $x=\left(x_{1}, \ldots , x_{m}\right) \in \mathbb {R}^{m}$ . Prove that there exist fractions $\frac{p_{1}}{q}, \ldots , \frac{p_{m}}{q}$ with arbitrarily large denominator $q \in \mathbb {N}$ ( $p_{1}, \ldots , p_{m} \in \mathbb {Z}$ ) such that

\left|x_{j}-\frac{p_{j}}{q}\right|<q^{-\left(1+\frac{1}{m}\right)} \quad \text{ for all } j=1,2, \ldots , m .

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Problem I.3.11

(a)(b)

Let $\alpha_{q}=\min_{p \in \mathbb {Z}}\left|\sqrt{2}-\frac{p}{q}\right|, q \in \mathbb {N}$ . Prove that

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(a)

$(2 q)^{-2}<\alpha_{q}<(2 q)^{-1}$ ;

(b)

if a strictly increasing sequence of indices $\left\{ q_{j}\right\}$ is such that $\alpha_{q_{j}} \leqslant \frac{C}{q_{j}^{2}}(j \in \mathbb {N})$ , where $C$ is a fixed number, then $\left\{ q_{j}\right\}$ grows at least like a geometric progression: $\frac{q_{j+1}}{q_{j}} \geqslant 1+\frac{1}{8 C}$ . Thus the lower bound in inequality a) is crude for most indices.

Problem I.3.12

Let $\alpha$ be an algebraic number of degree at most $r$ (i.e., $\alpha$ is a root of an algebraic polynomial of degree $r$ with integer coefficients). Prove Liouville's theorem: if $\alpha$ is irrational, then $\left|\alpha -\frac{p}{q}\right|>\frac{c_{\alpha }}{q^{r}}$ for all $p \in \mathbb {Z}$ and $q \in \mathbb {N}$ (the positive coefficient $c_{\alpha }$ depends only on $\alpha$ ).

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Remark.

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An irrational number $\alpha$ is called a Liouville number if it has arbitrarily good approximations by fractions: for any $r>0$ there exists a fraction $\frac{p}{q}(p \in \mathbb {Z}, q \in \mathbb {N}, q>1)$ such that $\left|\alpha -\frac{p}{q}\right|<\frac{1}{q^{r}}$ (for examples of such numbers see problem 3.19). Liouville's theorem immediately implies that $\alpha$ is a transcendental number.

Problem I.3.13

(a)(b)

An irrational number $\alpha$ is called a Liouville number if for any $r>0$ there exists a fraction $\frac{p}{q}(p \in \mathbb {Z}, q \in \mathbb {N}, q>1)$ such that $\left|\alpha -\frac{p}{q}\right|<\frac{1}{q^{r}}$ . Let $\alpha$ be an irrational non-Liouville number: $\left|\alpha -\frac{p}{q}\right| \geqslant \frac{c_{\alpha }}{q^{r}}$ for every fraction $\frac{p}{q}(p \in \mathbb {Z}, q \in \mathbb {N})$ , where $c_{\alpha }$ and $r$ are fixed positive numbers (this is possible only for $r \geqslant 2$ -- see 3.1).

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(a)

Generalize the result of problem 3.11 b) by proving that this bound is crude for most fractions $\frac{p}{q}$ : if $\left|\alpha -\frac{p_{j}}{q_{j}}\right| \leqslant \frac{C}{q_{j}^{r}}$ , where $p_{j} \in \mathbb {Z}, q_{j} \in \mathbb {N}$ and $q_{j+1}>q_{j}$ for all $j \in \mathbb {N}$ , then the sequence $\left\{ q_{j}\right\}$ grows no slower than a geometric progression (more precisely, $\frac{q_{j+1}}{q_{j}} \geqslant 1+\left(\frac{c_{\alpha }}{2 C}\right)^{\frac{1}{r-1}}$ ).

(b)

Estimate from below the denominators $q_{j}$ , if the sequence of fractions $\left\{ \frac{p_{j}}{q_{j}}\right\}$ is such that for all indices $j$

q_{j}<q_{j+1} \quad \text{ and } \quad \left|\alpha -\frac{p_{j}}{q_{j}}\right| \leqslant \frac{C}{q_{j}^{\rho }}

for some $C>0$ and $\rho \in (1, r)$ .

Note that, by Liouville's theorem, $\alpha$ can be taken to be any irrational algebraic number. Moreover, for $r>2$ almost all real numbers $\alpha$ satisfy the inequality $\left|\alpha -\frac{p}{q}\right| \geqslant \frac{c_{\alpha }}{q^{r}}$ for all fractions $\frac{p}{q}$ (see VIII.1.7).

Problem I.3.14

$\left\{ \varepsilon_{k}\right\}$ is an arbitrary sequence of $+1$ 's and $-1$ 's, and $\left\{ n_{k}\right\}$ is a strictly increasing sequence of natural numbers. Prove that the sum of the series $\sum \varepsilon_{k} 2^{-n_{k}}$ is irrational if $\overline{\lim }\left(n_{k+1}-n_{k}\right)=+\infty$ .

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Problem I.3.15

(a)(b)

$\left\{ \varepsilon_{k}\right\}$ is an arbitrary sequence of $+1$ 's and $-1$ 's, and $\left\{ n_{k}\right\}$ is a strictly increasing sequence of natural numbers. Let $\varlimsup \frac{n_{k+1}}{n_{1} n_{2} \ldots n_{k}}=+\infty$ . Prove that

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(a)

the sum of the series $\sum \frac{(-1)^{k}}{n_{k}}$ is irrational;

(b)

if $\frac{n_{k+1}}{n_{k}} \geqslant 2$ for all $k \in \mathbb {N}$ , then the sum of the series $\sum \frac{\varepsilon_{k}}{n_{k}}$ is irrational. Is this so in the case $\frac{n_{k+1}}{n_{k}} \geqslant Q>1$ ?

Problem I.3.16

(a)(b)

$\left\{ \varepsilon_{k}\right\}$ is an arbitrary sequence of $+1$ 's and $-1$ 's, and $\left\{ n_{k}\right\}$ is a strictly increasing sequence of natural numbers. Let $\varlimsup \frac{\ln n_{k}}{2^{k}}=+\infty$ . Prove that

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(a)

the sum of the series $\sum \frac{(-1)^{k}}{n_{k}}$ is irrational;

(b)

if $\frac{n_{k+1}}{n_{k}} \geqslant 2$ for all $k \in \mathbb {N}$ , then the sum of the series $\sum \frac{\varepsilon_{k}}{n_{k}}$ is irrational.

Problem I.3.17

(a)(b)

$\left\{ \varepsilon_{k}\right\}$ is an arbitrary sequence of $+1$ 's and $-1$ 's, and $\left\{ n_{k}\right\}$ is a strictly increasing sequence of natural numbers.

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(a)

Prove that the sum of the series $S=\sum \frac{\varepsilon_{k}}{F_{k}}$ is irrational, if $F_{k}=2^{2^{k}}+1$ are the Fermat numbers.

(b)

Let the sequence $\left\{ n_{k}\right\}$ be such that

n_{k+1} \geqslant n_{k}^{2}-n_{k}+1 \quad \text{ for all } \quad k \in \mathbb {N} .

Prove the rationality of the sum $\sum \frac{1}{n_{k}}$ , if $n_{k+1}=n_{k}^{2}-n_{k}+1$ for all sufficiently large $k$ .

Is the converse statement true? Is the number $\sum \frac{1}{2^{2^{k}}-2}$ rational?

Problem I.3.18

(a)(b)

$\left\{ \varepsilon_{k}\right\}$ is an arbitrary sequence of $+1$ 's and $-1$ 's, and $\left\{ n_{k}\right\}$ is a strictly increasing sequence of natural numbers. Let $\varlimsup \frac{\ln n_{k}}{3^{k}}=+\infty$ . Prove that

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(a)

the number $\sum \frac{(-1)^{k}}{n_{k}}$ is not a quadratic irrational;

(b)

if $\frac{n_{k+1}}{n_{k}} \geqslant 2$ for all $k \in \mathbb {N}$ , then the number $\sum \frac{\varepsilon_{k}}{n_{k}}$ is not a quadratic irrational.

Problem I.3.19

$\left\{ \varepsilon_{k}\right\}$ is an arbitrary sequence of $+1$ 's and $-1$ 's. Prove the transcendence of the numbers $\sum \varepsilon_{k} 2^{-k!}$ .

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Problem I.3.20

(a)(b)

$\left\{ \varepsilon_{k}\right\}$ is an arbitrary sequence of $+1$ 's and $-1$ 's, and $\left\{ n_{k}\right\}$ is a strictly increasing sequence of natural numbers. Let $\overline{\lim } \frac{\ln n_{k}}{k!}>0$ . Prove that

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(a)

the number $\sum \frac{(-1)^{k}}{n_{k}}$ is transcendental;

(b)

if $\frac{n_{k+1}}{n_{k}} \geqslant 2$ for all $k \in \mathbb {N}$ , then the number $\sum \frac{\varepsilon_{k}}{n_{k}}$ is transcendental.

Problem I.3.21

(a)(b)

Irrationality of the number $e$ . Using the expansions

e=\sum _{n \geqslant 0} \frac{1}{n!} \quad \text{ and } \quad \frac{1}{e}=\sum _{n \geqslant 0} \frac{(-1)^{n}}{n!},

prove that

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(a)

the number $e$ is irrational;

(b)

$A e^{2}+B e+C \neq 0$ , if $A, B, C$ are integers not all equal to zero.

Problem I.3.22

(a)(b)

Let $\sigma (n)$ be the sum of divisors of a natural number $n$ . Prove the irrationality of the sums of the series

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(a)

$\sum \frac{\sigma (n)}{n!}$ ;

(b)

$\sum (-1)^{n} \frac{\sigma (n)}{n!}$ .

Problem I.3.23

(a)(b)

Irrationality of the number $\pi$ .

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(a)

Let $H_{n}=\frac{1}{n!} \int_{-\pi / 2}^{\pi / 2}\left(\frac{\pi^{2}}{4}-t^{2}\right)^{n} \cos t d t(n=0,1,2, \ldots )$ . Prove that $H_{n}=P_{n}\left(\pi^{2}\right)$ , where $P_{n}$ is an algebraic polynomial of degree at most $n$ with integer coefficients.

(b)

Deduce from this that $\pi^{2}$ (and hence also $\pi$ ) is an irrational number.

Problem I.3.24

(a)(b)

Irrationality of the values of the exponential function at rational points.

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(a)

Let $J_{n}(x)=\frac{1}{n!} \int_{-x}^{x}\left(x^{2}-t^{2}\right)^{n} e^{t} d t \quad (n=0,1,2, \ldots )$ . Prove that $J_{n}(x)=A_{n}(x) e^{x}+B_{n}(x) e^{-x}$ , where $A_{n}, B_{n}$ are algebraic polynomials of degree at most $n$ with integer coefficients.

(b)

Deduce from this that $e^{r} \notin \mathbb {Q}$ , if $r \in \mathbb {Q}, r \neq 0$ .

Problem I.3.25

(a)(b)

Irrationality of the values of the tangent at rational points.

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(a)

Let $H_{n}(x)=\frac{1}{n!} \int_{-x}^{x}\left(x^{2}-t^{2}\right)^{n} \cos t d t(n=0,1,2, \ldots )$ . Prove that $H_{n}(x)=C_{n}(x) \cos x+S_{n}(x) \sin x$ , where $C_{n}, S_{n}$ are algebraic polynomials of degree at most $n$ with integer coefficients.

(b)

Deduce from this that $\operatorname {tg} r \notin \mathbb {Q}$ , if $r \in \mathbb {Q}, r \neq 0$ .