10 problems

Let $n$ be an integer with $n\ge 9$, and consider the base-$3$ expansion of $2^n$. Erdős's ternary conjecture. For every $n\ge 9$, this expansion contains the digit $2$. This class…

Let … mu$. Consider the sequence$({x^n}){ngeq 1}$modulo one. **Baker's conjecture.** For every such self-similar measure, the sequence$({x^n}){ngeq 1}…

Let $f(x,y)=(y^2-x^3)/x$. For $Q\in\mathbb{N}_{\geq 2}$, let $C_Q$ be the set defined in the paper for this rational function and quotient $Q$. Nonemptiness and infinitude conjectu…

For a real number $\alpha>1$, write $\{\alpha^n\}$ for the fractional part of $\alpha^n$. The sequence $(\{\alpha^n\})_{n\ge1}$ has Poissonian $k$-point correlation when its $k$-po…

Type conjecture. If $pq$ is even, then there exist $k$ and $n$ such that

Let $T$ be the sequence generated by the $6$-GP-free process, and interpret a gap of size $O((\log x)^C)$ as the existence of a constant implicit in the big-$O$ bound, independent…

Let $N$ be a modulus, let $a>1$ be a base, and for each starting point $A$ with $0<A<N$ let $\lambda_n(A)$ be the Kolmogorov stochasticity parameter of the residues modulo $N$ of t…

Let $N$ be a modulus, and let $\lambda_n$ denote the Kolmogorov stochasticity parameter of the residues modulo $N$ of the first $n$ terms of a geometric progression with ratio…

Rankin's set conjecture. Rankin's set $G^*_3$ has the largest possible density among geometric-progression-free sets which have a density, so

Let $\xi>0$ be fixed, let $x>1$, and write $\{\xi x^n\}$ for the fractional parts of the geometric progression. Let $D_N$ denote the discrepancy of the first $N$ terms…