Typical group automorphisms

Definitions and mathematical statement

For each prime $p$, independently choose a Bernoulli random variable $\xi_p\in\{0,1\}$ with probability $1/2$ of either value, and set $Q=\{p:\xi_p=1\}$. For a positive integer $a$, let $v_p(a)$ be the largest integer $j\ge0$ such that $p^j$ divides $a$, and put $|a|_p=p^{-v_p(a)}$. Define \[ F_Q(n)=(2^n-1)\prod_{p\in Q}|2^n-1|_p =\prod_{p\notin Q}p^{v_p(2^n-1)}. \] The products are finite in effect, because only prime divisors of $2^n-1$ contribute a nonunit factor. The intended periodic-point growth question is whether \[ \mathbb P\!\left(\limsup_{n\to\infty}\frac{\log F_Q(n)}{n} =\log2\right)=1. \] The same random set $Q$ is used for every $n$. All logarithms are real natural logarithms. Since $1\le F_Q(n)\le2^n-1$, the claimed value is the largest possible exponential upper growth rate.

Parentheses in the formula matter: the logarithm applies to the entire product defining $F_Q(n)$, as required by the periodic-point interpretation and the source's lower bound. This is stronger than computing an expectation for each $n$: almost every single infinite prime selection must have a subsequence along which its retained prime powers recover the full exponential growth rate. No independence between the numbers $F_Q(n)$ for different $n$ is assumed. In particular, repeating the prime selection afresh for each index would define a different probability problem.

Short English statement

For almost every independent half-selection of primes, do the surviving factors of $2^n-1$ retain the full exponential limsup growth?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4300004 (AMR-042-0004), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Ward - Six problems in Algebraic Dynamics (2006), Problem D. Original source indexed by AMR-042-0004. https://www.imath.kiev.ua/~skolyada/kevin.pdf.