Density of sets of integers

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Problem II.2.14 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section II.2 "Averaging of Sequences"

(a)(b)

Let $E=\left\{ n_{1}, n_{2}, \ldots \right\}$ , where $n_{1}<n_{2}<\ldots$ , be an infinite subset of $\mathbb {N}$ , and let $E_{m}=\left\{ n \in E \mid n \leqslant m\right\}$ . If the limit $\theta (E)=\lim \frac{1}{m} \operatorname {card}\left(E_{m}\right)$ exists, it is called the density of the set $E$ (or the density of the sequence $\left\{ n_{k}\right\}$ ). Prove that

(a)

$\theta (E)=\lim \frac{k}{n_{k}}$ ;

(b)

for any number $\alpha \in [0,1]$ there exists a set $E \subset \mathbb {N}$ such that $\theta (E)=\alpha$ .

Problem II.2.15 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section II.2 "Averaging of Sequences"

(a)(b)(c)

Let $\left\{ x_{n}\right\}$ be a sequence of real numbers. Prove that

(a)

if $x_{n} \rightarrow a \in \mathbb {R}$ , then $\frac{1}{n} \sum_{k=1}^{n}\left|x_{k}-a\right| \rightarrow 0$ ;

(b)

if the relation $\frac{1}{n} \sum_{k=1}^{n}\left|x_{k}-a\right| \rightarrow 0$ holds for some $a \in \mathbb {R}$ , then there exists a sequence $\left\{ n_{k}\right\}$ of density 1 (see Problem 2.14) such that $x_{n_{k}} \rightarrow a$ ;

(c)

if the sequence $\left\{ x_{n}\right\}$ is bounded and $x_{n_{k}} \rightarrow a$ for some sequence $\left\{ n_{k}\right\}$ of density 1, then $\frac{1}{n} \sum_{k=1}^{n}\left|x_{k}-a\right| \rightarrow 0$ .

Problem II.2.17 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section II.2 "Averaging of Sequences"

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

We call the periodic density of a set $E \subset \mathbb {N}$ the number $\underline{\lim } \frac{1}{m} \operatorname {card}\left(\widetilde{E}_{m}\right)$ , where

\widetilde{E}_{m}=\left\{ n \bmod m \mid n \in E\right\}

is the set of all possible remainders of numbers from $E$ upon division by $m$ .

(a)

Prove that if the set $E$ has a density, then it does not exceed its periodic density.

(b)

Prove that a set with zero periodic density has zero density.

(c)

Give an example of a set with zero density whose periodic density equals one.

(d)

What is the periodic density of the set of all natural numbers representable as a sum of distinct terms of the form $\pm 4^{k}, k \in \mathbb {Z}_{+}$ ?

Problem II.2.18 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section II.2 "Averaging of Sequences"

Prove that the periodic density of the set of prime numbers equals zero, and hence (see 2.17 b)), for $\pi (n)$ (the number of primes not exceeding $n$ ) the estimate $\pi (n)=o(n)$ holds (a more precise result is contained in Problem I.2.13).

Problem II.2.19 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section II.2 "Averaging of Sequences"

Let $E \subset \mathbb {R}$ ,

\begin{aligned} & n_{+}(r)=\sup _{x \in \mathbb {R}} \operatorname {card}(E \cap [x, x+r]), \\ & n_{-}(r)=\inf _{x \in \mathbb {R}} \operatorname {card}(E \cap [x, x+r])(r>0) . \end{aligned}

Prove that if $n_{+}\left(r_{0}\right)<\infty$ for some $r_{0}>0$ , then finite limits $d_{ \pm }=\lim_{r \rightarrow +\infty } \frac{n_{ \pm }(r)}{r}$ exist (the upper and lower densities of the set $E$ ).

References

[МГЛ+04] Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. Избранные задачи по вещественному анализу. 2-е изд., Невский Диалект; БХВ-Петербург, 2004.