8 problems

Let $\alpha\in\mathbb{R}$ be an irrational algebraic number. A real number is normal if its digit frequencies are uniform in every integer base. Normality conjecture for irrational…

Let $L$ be a number field and let $\alpha\in L$ be irrational. Let $\mathfrak{a}$ be an unramified ideal of degree one that is prime to the denominator of the principal ideal…

Let $\alpha$ be an irrational algebraic number and let $n>1$ be an integer. An $n$-adic expansion of $\alpha$ is a representation $\sum_{l=0}^{\infty}a_ln^l$ with $0\le a_l<n$ sati…

Let $\{\varphi_i(x)=\lambda_i x+t_i\}_{i\in \mathcal{I}}$ be an iterated function system of contracting similarities, and let $b\geq 2$ be an integer. Suppose that … for some…

Let $b\geq 2$ be an integer base, and let $(\alpha_1,\ldots,\alpha_n)$ be an $n$-tuple of real numbers. Writing their base-$b$ expansions in aligned columns, call the tuple normal…

Let $m\ge 2$ be an integer, and consider the digits in basis $m$ of the fractional part of either Pi or the Euler constant. Denote by $(\mathcal P)$ the large-deviation property de…

Maximal complexity conjecture. The complexity of $\alpha$ satisfies

Let $x$ be a real irrational algebraic number and let $g\geqslant2$ be a positive integer. A real number is normal in base $g$ if it is simply normal in base $g^m$ for every…