9 problems

Fix an integer $k\geq2$. In the discrete stick-fragmentation process, break each stick into $k$ pieces by choosing $k-1$ cut points recursively according to the uniform distributio…

Fix an even modulus $n\geq2$, a subset $S\subset\{0,\dots,n-1\}$ of residue classes, and the associated discrete stick-fragmentation process with stopping set determined by $S$. Le…

Let $\mathfrak{S}\subseteq\mathbb{Z}_+$ be a stopping set such that the natural density exists, and write … Assume that $r>0$. Density-half conjecture. The set of dead stick length…

Lower-dimensional volume conjecture. A $d$-dimensional volume for an $m$-dimensional object, where $d<m$, undergoing a fragmentation process follows the Benford distribution for th…

Rectangle-perimeter conjecture. The perimeters of the sub-rectangles from a rectangle undergoing an unrestricted fragmentation process converge to Benford behavior.

Let $a\geq 2$ be an integer that is not a power of $10$, let $d\in\{1,2,\dots,9\}$, and let $E_d(N,\{a^n\})$ denote the Benford error. Nonnormal distribution conjecture for Benford…

Let $a>0$ be a real number satisfying the relevant condition, and write $\alpha=\log_{10}a$. Suppose that the continued fraction expansion of $\alpha$ satisfies … For a digit…

Fix a monotonically increasing sequence $\{a_n\}$. In the additive decomposition process, a stick whose length belongs to $\{a_n\}$ does not decompose further; sticks of length $1$…

Consider an additive decomposition process starting from a stick of integer length $L$: choose uniformly an integer cut point $c\in[1,L-1]$, replace the stick by pieces of lengths…