16 problems

Let … where $a_n=(-1)^{s_2(n)}\in\{-1,1\}$ and $s_2(n)$ is the sum of the binary digits of $n$. Gawron–Miska–Ulas conjecture. For every integer $m\geq 2$, the coefficient sequence…

Let $t(n)$ denote the Thue–Morse sequence, and let $P(x)$ be a non-negative integer-valued polynomial. Gelfond's conjecture. The values of $t(P(n))$ are equidistributed between $0$…

Correlation-measure conjecture for the square subsequence.

Let the lower bounds in Proposition denote the stated lower bounds for the relevant monochromatic arithmetic-progression parameters in the Thue–Morse sequence. Exactness conjecture…

Let ${\bf t}=t_0t_1t_2\cdots$ be the Thue–Morse word, and let $A$ denote an occurrence of the factor ${\tt 01101}$ in ${\bf t}$ and $B$ an occurrence of its complementary factor…

Sixfold strict increase conjectures. The sequences $r_6(n)$ and $s_6(n)$ satisfy

For a sequence $\mathcal{S}$, let a step be an integer $N$ at which its maximum order complexity changes from $M(\mathcal{S},N-1)$ to $M(\mathcal{S},N)$, and define the ratio of su…

Let $\mathcal{T}_P$ denote the Thue–Morse sequence along a polynomial subsequence, where $P$ is a polynomial of degree $d$. Write $M(\mathcal{T}_P,N)$ for its maximum order complex…

Let $(t_n)$ be the Thue–Morse sequence, and consider the subsequence $(t_{3^n})$. A finite binary word is called a pattern in this sequence if it is a finite block of consecutive t…

For each positive integer $k$, let … where … and $t(m)\in\{\pm1\}$ is the parity of the sum of the base-$2$ digits of $m$. For every $k>0$, write … for suitable constants $C_k>0$ a…

Let ${\cal T}'=(t_{n^2})_{n\geq 0}$ be the subsequence of the Thue–Morse sequence along the squares, and let $M({\cal T}',N)$ denote its $N$th maximum order complexity. The notatio…

Let $\mathbf{t}=t_0t_1t_2\cdots$ be the Thue–Morse word. The circular critical exponent of a factor is the largest exponent of a repetition occurring when the factor is viewed as a…

Consider an $\mathcal N=4$ quiver with total gauge group $G\cong U(1)^\alpha$ and a decoupled overall $U(1)$ factor. Let $J$ and $K$ be the index sets of thimble contributions, let…

Let $\mathbf{t}=t_0t_1t_2\cdots$ be the Thue–Morse sequence, defined by $t_0=0$, $t_{2n}=t_n$, and $t_{2n+1}=1-t_n$ for $n\geq 0$. Let $f(n)$ denote the number of unbordered factor…

Let $b,r$ be positive integers with $\gcd(b-1,r)=1$, and let $c$ be any integer. Write $s_b(n)$ for the sum of the base-$b$ digits of $n$. Bounded-offset conjecture. There exists a…

Periodicity conjecture. The sequence $\{\gamma_a(n)\}$ is periodic. If $2^m\mathrel{\|}a$, then its minimal period has $2^{m+1}$ terms. If $a$ is evil, this period consists of the…