14 problems

Let $R$ be the set of Kolmogorov-random binary strings defined using the fixed universal Turing machine $U$, and let $k$ be the threshold supplied by Chaitin's incompleteness theor…

Existence conjecture. There is a highly normal number that is not KL-stochastic.

Let $\mathbb A^\infty$ be the space of one-sided infinite sequences over the alphabet $A$, with shift operator $T$. Let $\mu$ be a computable ergodic $T$-invariant measure, and def…

Quantum ShannonfMcMillanfBreiman conjecture. If $\mu$ is an ergodic computable state on $M_\e1$ and $\rho$ is quantum Martin-Lf random with respect to $\mu$, then

Let $(X,\mu)$ be a computable probability space. Let $T_1,\ldots,T_k$ be computable measure-preserving transformations that commute pairwise. Let $P$ be a $\Pi^0_1$ class with…

Assume that $\mathbb{X}$ and $\mathbb{Y}$ are computable metric spaces, that $\{\mu_a\}_{a\in\mathbb{N}^{\mathbb{N}}}$ is a uniformly computable family of probability measures on…

Let $A_N$ denote nondeterministic automatic complexity. For a random infinite binary sequence, let $D_n$ be the nondeterministic automatic complexity deficiency of its length-$n$ p…

Let $(a_i)_{i\in\mathbb{N}}$ be a sequence of distinct integers such that the function mapping the unary input $1^j$ to $a_j$ is polynomial-time computable. For a real $y$, write…

A real $x$ is polynomially random if it satisfies the polynomial-time randomness notion used in the source, and it is rationally normal if it is normal in every rational base $r>1$…

Let $\omega\in 2^{\mathbb{N}\times\mathbb{N}}$. A real is Doob random if it satisfies the paper's characterization in terms of computable martingales, and it is e.c.u. random if it…

A real is strongly $\gamma$-random if it is strongly random with parameter $\gamma$, and Hippocrates $\gamma$-energy random if it satisfies the Hippocrates energy-randomness notion wi…

Countability conjecture. There are only countably many id-jump traceable reals.

Let $A,B esubset \mathbb R+$ be sets of nonnegative real numbers. The set $B$ anticipates $A$ when the regular gamblers' wagers from $B$ can be evaded against gambler 0's wagers…

Let $\alpha$ be a computable limit ordinal, possibly with additional closure properties such as closure under addition. For an oracle $A$, let…