7 problems

Uniqueness conjecture. For any self-similar crystallographic or self-affine tile, the combinatorial structure of the neighbor automaton uniquely determines the iterated function sy…

A rational number $p/q$ has an antipalindromic quotient representation if there are antipalindromic numbers $A$ and $B$ such that $p/q=A/B$. Rational antipalindromic quotient conje…

Let $p>1$ be odd. A rational number $p/q$ has a palindromic quotient representation if there are palindromic numbers $A$ and $B$ such that $p/q=A/B$. Odd rational palindrome quotie…

Let $(W,S)$ be a Coxeter system, let $\Gamma$ be the set of gates of $W$, and let joins be taken in the weak order on $W$. A Garside shadow conjecture. The set $\Gamma$ is closed u…

Let ${\mathrm{TB}}$ be the total budget, and let a unitary game be a game in the class defined by the paper. Associate to it the corresponding 0-bidding automaton $\mathcal{A}$, wh…

Let $(W,S)$ be an irreducible Coxeter system. Let $\mathcal A_0(W,S)$ denote the canonical automaton recognizing the language of reduced words, let $\Sigma$ denote the set of small…

Let $(W,S)$ be a Coxeter system, let $\tilde S$ be its smallest Garside shadow, and let $\mathcal A_{\tilde S}(W,S)$ be the associated finite deterministic automaton. Write…