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Soprunov and Soprunova introduced the notion of a good infinite family of toric codes. We prove that such good families do not exist by proving a more general Szemer\'edi-type result: for all $c\in(0,1]$ and all positive integers $N$, subsets of density at least $c$ in $\{0,1,\dots,N-1\}^n$ contain hypercubes of arbitrarily large dimension as $n$ grows., Comment: 10 pages. Comments welcome

We show that every Sturmian word has the property that the distance between consecutive ending positions of cubes occurring in the word is always bounded by $10$ and this bound is optimal, extending a result of Rampersad, who proved that the bound $9$ holds for the Fibonacci word. We then give a general result showing that for every $e \in [1,(5+\sqrt{5})/2)$ there is a natural number $N$, depending only on $e$, such that every Sturmian word has the property that the distance between consecutive ending positions of $e$-powers occurring in the word is uniformly bounded by $N$.

We study point modules of monomial algebras associated with symbolic dynamical systems, parametrized by proalgebraic varieties which 'linearize' the underlying dynamical systems. Faithful point modules correspond to transitive sub-systems, equivalently, to monomial algebras associated with infinite words. In particular, we prove that the space of point modules of every prime monomial algebra with Hilbert series $1/(1-t)^2$ -- which is thus thought of as a 'monomial $\mathbb{P}^1$' -- is isomorphic to a union of a classical projective line with a Cantor set. While there is a continuum of monomial $\mathbb{P}^1$'s with non-equivalent graded module categories, they all share isomorphic parametrizing spaces of point modules. In contrast, free algebras are geometrically rigid, and are characterized up to isomorphism from their spaces of point modules. Furthermore, we derive enumerative and ring-theoretic consequences from our analysis. In particular, we show that the formal power series counting the irreducible components of the moduli schemes of truncated point modules of finitely presented monomial algebras are rational functions, and classify isomorphisms and automorphisms of projectively simple monomial algebras.