13 problems

Let $M\in M_2(\mathbb{Z})$ be an expanding matrix and let $D=\{(0,0)^t,(\alpha_1,\alpha_2)^t,(\beta_1,\beta_2)^t\}$ satisfy $\alpha_1\beta_2-\alpha_2\beta_1=0$. Then $D$ is colline…

Let $M\in M_n(\mathbb{Z})$ be an expanding matrix and let $D\subset\mathbb{Z}^n$ be a finite set. A Hadamard triple is a triple $(M,D,S)$, where $S\subset\mathbb{Z}^n$ is finite wi…

Let $(T_1,\ldots,T_N)$ be an affine iterated function system acting on $\mathbb{R}^d$, and let $\operatorname{dim_{\mathsf{aff}}}(T_1,\ldots,T_N)$ denote its affinity dimension. A…

Let the self-affine measure and its linear part satisfy the condition of Theorem, except possibly the assumptions that the Zariski closure is $\mathbb{R}$-splitting and that the li…

Let $T_1,\dots,T_N$ be contracting non-singular $2\times 2$ upper triangular matrices, let $c_i,d_i$ be the diagonal entries of $T_i$, and let $\mu$ be a planar self-affine measure…

Let $R$ be an expansive integer matrix, let $B$ be a finite set of integer vectors with $\#B=N$, and let $\mu(R,B)$ be the equal-weight self-affine probability measure associated w…

Let $s_r$ be the unique solution of … For the self-affine measure $\mu$ on the Bedford–McMullen carpet $E$, consider the two conditions: (a)…

Let $M\in M_n(\mathbb{Z})$ be an expanding matrix, and let $\mathcal{D},S\subset\mathbb{Z}^n$ be finite sets with the same cardinality $q$. The triple $(M,\mathcal{D},S)$ is a Hada…

Fix real parameters $p\in(0,1)$ and $q>1$, and write $\mathfrak{r}_q(A_1,A_2,p)$ for the generalized Rényi dimension associated with the probability vector $(p,1-p)$. Discontinuity…

Let $\Phi=\{f_i(\underline{x})=A_i\underline{x}+\underline{t}_i\}_{i=1}^N$ be an affine iterated function system on the plane whose attractor contains at least two points and which…

Fix positive integers $m,n$ with $2\leq m\leq n$, a digit set $G\subset\{0,1,\ldots,n-1\}\times\{0,1,\ldots,m-1\}$ with $N=\operatorname{card}(G)\geq2$, affine maps…

Negative-$q$ spectrum conjecture. The $L^q$-spectra of graph-directed self-similar measures exist for every $q\in\mathbb{R}$. Moreover, for $q<0$ and measures satisfying the ROSC,

Self-affine Fourier-frame conjecture. The only absolutely continuous self-affine measures on $\mathbb{R}^n$ admitting a Fourier frame are characteristic functions of self-affine ti…