Open-problem statements, with their sources attached.

Problem 10.1. Are there a transcendental number $\alpha$ and a positive real number $\xi$ such that $\lVert \xi \alpha^n \rVert$ tends to~$0$ as~$n$ tends to infinity? [Har19] (Trivial for $|\alpha| < 1$)

Problem 10.2. To prove that $\lVert e^n \rVert$ does not tend to 0 as n tends to infinity.

Problem 10.3. To prove that there exists a positive real number~$c$ such that $\lVert e^n \rVert > e^{−cn}$, for every~$n \ge 1$. Posed by Mahler [Mah53].

Waldschmidt [Wal03] conjectured that a stronger result holds, namely that there exists a positive real number~$c$ such that $\lVert e^n \rVert > n^{-c}$ for every~$n \ge 2$. This is supported by metrical results [Kok45].

Note: the bound $n^{-c}$ equals $1$ when $n = 1$ for all $c$, while the distance to the nearest integer is always at most $1/2$, so the conjecture must start at $n \ge 2$.

Problem 10.4. Let $\xi$ be a non-zero real number and $\alpha > 1$ be a real number. The spectrum of the sequence $(\xi \alpha^n)_{n \ge 1}$ is at most countable. Posed by Mendès France [Men73].

Problem 10.5 (first part). Let $\mathbb{K}$ be a real number field. Then, for any $\varepsilon > 0$, there exists a lacunary sequence $(t_n)_{n \ge 1}$ of positive numbers in $\mathbb{K}$ such that $\limsup_{n \to \infty} \{\xi t_n\} \ge 1 - \varepsilon,$ for any real number $\xi$ not in $\mathbb{K}$.

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.