Open-problem statements, with their sources attached.

Problem 10.5 ("moreover" clause). With the same hypotheses as problem_10_5, the sequence $(t_n)$ can be chosen so that, for any real $\xi$ not in $\mathbb{K}$, each subinterval of $[0, 1]$ of length $\varepsilon$ contains a limit point of the sequence $(\{\xi t_n\})_{n \ge 1}$. This is strictly stronger than problem_10_5: the limsup bound is the special case at the subinterval $[1 - \varepsilon, 1]$.

Problem 10.6. Find a very rapidly increasing sequence $(m_n)_{n \ge 1}$ of positive integers such that $(\{\xi m_n\})_{n \ge 1}$ is dense modulo one for every irrational number $\xi$. Note: Furstenberg's $2^m3^n$ is sublacunary but requires two parameters.

Problem 10.6, intermediate-growth variant.

Problem 10.7. Let $\varepsilon$ be a positive real number. Are there arbitrarily large real numbers $\alpha$ such that $\alpha$ is not a Pisot number and all the fractional parts $\{\alpha^n\}$, $n \ge 1$, are lying in an interval of length $\varepsilon / \alpha$? [Bug12b]

Problem 10.8 ($p$-adic Littlewood conjecture). For every real number $\xi$ and every prime number $p$, $\inf_{q \ge 1} q \cdot \lVert q \xi \rVert \cdot |q|_p = 0,$ where $\lVert \cdot \rVert$ denotes the distance to the nearest integer and $|\cdot|_p$ denotes the $p$-adic absolute value. Posed by de Mathan and Teulié [dMT04].

The sequence (3/2)^n is equidistributed modulo 1.

For any transcendental number x, the sequence x * (3 / 2) ^ n is equidistributed modulo 1.

Find an accumulation point of the sequence (3/2)^n modulo 1.

For any $0<\alpha<1$, let $f(\alpha,n)=\frac{1}{\log n}\sum_{1\leq k\leq n}(\tfrac{1}{2}- \{ \alpha k\})$. Does $f(\alpha,n)$ have an asymptotic distribution function?

In other words, is there a non-decreasing function $g$ such that $g(-\infty)=0$, $g(\infty)=1$, and $\lim_{n\to \infty}\lvert \{ \alpha\in (0,1): f(\alpha,n)\leq c\}\rvert=g(c)$?

Are there infinitely many solutions to $\phi(n) = \phi(n+1)$, where $\phi$ is the Euler totient function?

Erdős [Er85e] says that, presumably, for every $k \geq 1$ the equation $\phi(n) = \phi(n+1) = \cdots = \phi (n+k)$ has infinitely many solutions.

[Er85e] Erdős, P., Some problems and results in number theory. Number theory and combinatorics. Japan 1984 (Tokyo, Okayama and Kyoto, 1984) (1985), 65-87.

For any fixed c > 0, if x is sufficiently large then there exists n ≤ x such that the values of φ(n+k) are all distinct for 1 ≤ k ≤ (log x)^c. This is an open problem.

A general version asks, for a fixed $r \in \mathbb{N}$, if a set $A \subseteq \{1, ..., N\}$ has no $a \in A$ and $b_1, ..., b_r \in A$ such that $a | (b_1 + ... + b_r)$ and $a < \min(b_1, ..., b_r)$, then is it true that $|A| \le N/(r+1) + O(1)$?

Let $t>1$ be a rational number. Is $\sum_{n=1}^\infty\frac{1}{t^n-1}=\sum_{n=1}^\infty \frac{\tau(n)}{t^n}$ irrational, where $\tau(n)$ counts the divisors of $n$?

A conjecture of Chowla.

Let $k≥3$. Are there $k$ consecutive primes in arithmetic progression?

Are there only finitely many unitary perfect numbers?

Are there $11$ consecutive primes in arithmetic progression?

Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Is it true that $f(n)=o(n)$?

It is open, even for $k=3$, whether there are infinitely many such progressions.

Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Is it true that $f(n)=o(n)$ for almost all $n$?