12 problems

Asymptotic conjecture for $(2,k,k)$-numbers. For each $k$,

Heuristic exponent conjecture for $(2,1,2)$-numbers.

Let $d\in\mathbb{N}$, and let an order-2 digital sequence mean a digital sequence over the finite field $\mathbb{F}_2$ with order parameter $2$. For an infinite sequence…

Let $\left(\mathbf{H}_n\right)_{n\ge0}$ be the hybrid sequence from the paper's main theorem, and let $D_{N,\mathbf{H}}$ denote its discrepancy. Lower-bound conjecture for the main…

Let $\Theta(t)\in\mathbb{F}_q((t^{-1}))$ be a Laurent series, let $k$ be a natural number, and let $P_1(t),\dots,P_k(t)\in\mathbb{F}_q[t]$ be coprime irreducible polynomials. Assum…

Let $P(t)\in\mathbb{F}_q[t]$ be an irreducible polynomial, and let $\Theta(t)\in\textbf{Bad}(P(t),q)$. Let $K_n(\Theta)$ denote the digital Kronecker sequence associated to…

Let $\varphi=(1+\sqrt{5})/2$ be the golden ratio, let $s_Z(n)$ denote the sum of digits of $n$ in Zeckendorf base, and let $\mathcal{S}_Z=(s_Z(n)\bmod 2)_n$. Write…

Let $s\in\mathbb{N}$ and let $\mathcal{S}$ be a sequence in $[0,1)^s$. Write $D_N^{\ast}(\mathcal{S})$ for its star discrepancy at $N$ points. Grand discrepancy conjecture. For eve…

Let $X^C$ be a digital NUT sequence, and let $L_p(X_N^C)$ denote its $L_p$ discrepancy at the first $N$ points. Here $p$ is as in the discrepancy setting of the paper. Lower-bound…

Let $T=(T(n))_{n\ge 0}$ be the Tribonacci numeration system built on the language of words over $\{0,1\}$ avoiding three consecutive ones, and let $A_T$ be the corresponding summat…

For an integer base $k\ge 3$, let $A_k(n)$ be the analogue of $A(n)$ obtained from words and subwords in the language $\{1,\ldots,k-1\}\{0,\ldots,k-1\}^*\cup\{\varepsilon\}$. Base-…

Let the explicit point sets and sequences constructed in this paper be given, and let ${\cal L}_\infty$ denote their $\mathcal{L}_\infty$ discrepancy. Optimal-rate conjecture. The…