10 problems

Quantitative Erdős–Kac conjecture for Beatty sequences. The distance between $X_N$ and the standard Gaussian $\mathcal{N}(0,1)$ is bounded above by

For an integer $j\geq 1$, let $\alpha_j$ be the $j$-th metallic mean, so that $\alpha_j^2=j\alpha_j+1$. Let $r_1$ be the unique positive real number such that all terms defined by…

Frantzikinakis's conjecture. For any such multiplicative functions, one has

Duchêne–Rigo conjecture. There exists an invariant game having

Fraenkel's conjecture. If the system has no multiplicity, then $\alpha_i=p/q_i$, where

Let $d\geq 3$, let $\alpha_1<\alpha_2<\ldots<\alpha_d$, and let $\gamma_1,\ldots,\gamma_d$ be real numbers. Suppose that the vectors … for $n\in\mathbb{N}$, split the positive inte…

Let $a>1$ and $b>1$ be irrational numbers. The Beatty-sequence logarithm bound by $1$. … Numerical data in the source suggest this stronger bound than the preceding conjecture. The…

Let $a>1$ and $b>1$ be irrational numbers. The Beatty-sequence logarithm bound. … This conjecture is motivated by estimating the corresponding series through the quantities…

Beatty-periodicity conjecture. The set $\mathcal{P}(G)$ is periodic if and only if the modulus $\alpha$ of $(a_n)$ is rational. This proposes a complete rational-versus-irrational…

Let $\alpha>0$ be an irrational number and let $\beta$ be a real number. The expression $\floor{\alpha p+\beta}$ denotes the largest integer not exceeding $\alpha p+\beta$. Primes…