5 problems

Rampersad–Shallit–Vandomme conjecture. For every alphabet size $d\geq 5$, the minimal critical exponent among balanced sequences over a $d$-letter alphabet equals

Let $k\geq 3$, and let $\{r_1,\dots,r_k\}$ be a set of distinct positive reals with $r_1>\dots>r_k>0$. A sequence over a $k$-element set is balanced if, for every symbol, the numbe…

Optimality conjecture. The bound is attained for infinitely many even values of $d$; that is, equality with the refined upper bound is expected for infinitely many even $d$.

New conjecture. For every alphabet size $d\geq 11$, the minimal critical exponent of a balanced sequence over a $d$-letter alphabet equals

Let $\mathcal{B}=\\{s\in\\{\mathtt{0},\mathtt{1}\\}^{\mathbb{Z}}:s\text{ is balanced}\\}$ be the set of all balanced bi-infinite binary sequences, and let … be their language of fi…