5 problems

Sturmian-region conjectures. (i) If $-4\leq r-s\leq3$, then, except for $(r,s)=(4,1)$, $w_{r,s}$ is Sturmian. (ii) If $r-s=4$, then $w_{r,s}$ is Sturmian if and only if $r$ is even…

Consider the rotor-router growth model on $\mathbb{Z}^d$, and let $A$ be its occupied region after particles are added at the origin. Rotor-router Euclidean-ball conjecture. The li…

Let $a,b,c$ be a positive Pythagorean triple, so that … In the one-dimensional rotor-router model, start with $A=[-a,0]$, use the case $r=s=1$, and measure time in iterations. Prop…

Propp's conjecture. The quantity $\mu(t)$ is bounded independently of $t$. The paper proves this conjecture and further confines the limit points of the sequence $\mu$ to an interv…

Consider the deterministic rotor-router growth model on $\mathbb{Z}^d$, and compare its asymptotic occupied regions with those of internal diffusion-limited aggregation (IDLA). Rot…