I recently asked in [this thread](https://mathoverflow.net/questions/434320/lower-bounds-for-pattern-complexity-of-aperiodic-subshifts) about lower bounds on the complexity in the case where we have an aperiodic subshift. If I denote $c\_n(\Omega)$ as the number of possible patterns on $Q\_n=\{0,...,n−1\}^d$ and had hoped for estimates of the form


$$ \liminf\_{n\to \infty} \frac{c\_n(\Omega)}{n^d}\geq C\_d. $$


In the same thread, Ville Salo pointed to a work by Julien Cassaigne showing this is not the case when $d=3$. However, this question was motivated by a more specific class of **linearly repetitive** subshifts. Since the counterexample was motivated by a conjecture by Lagarias and Pleasents, [Repetitive Delone sets and quasicrystals](https://arxiv.org/abs/math/9909033), I noticed that the same paper references that this result is true when $\Omega$ is linearly repetitive. This is stated in Conjecture 1.2b and is said to be proven by Lenz in [Aperiodic linearly repetitive Delone sets are densely repetitive](https://arxiv.org/abs/math/0208132).


However I wished to verify whether this is me misinterpreting the result stated, as this is stated in the language of Delone sets and I am interested in such a result for subshifts. It seems like the difference in setting isn't important for Cassaigne's result in the [counterexample paper](https://oleron.sciencesconf.org/conference/oleron/pages/pleasants.pdf), but I'm not sure why that is necessarily so? I'm wondering whether the transitions between the two settings, might not mean that the result by Lenz is valid in subshifts?



Lower bounds for pattern complexity of linearly repetitive aperiodic subshifts
Just to give closure to this question, I think an earlier paper by Boris Solomyak, [Nonperiodicity implies unique composition for self-similar translationally finite Tilings](https://link.springer.com/article/10.1007/PL00009386), actually explicitly solves my question in the desired setting. In the paper by David Lenz, he shows the property of the complexity function in *Lemma 2.2* . This lemma follows easily from Lenz's *Lemma 2.1*, which states that a return vector that is 'too small' must be a period of the minimal subshift. Lenz mentions that *Lemma 2.1* is proven similarly to *Lemma 2.4* in Solomyak's paper.


In Solomyak's paper, dealing with self similar tilings with finite local complexity, the aforementioned lemma is phrased as follows.



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> Let $\mathcal{U}$ be a self similar tiling with translation symmetry group $\mathcal{K}$. There exists a constant $N\geq 1$ such that, for any $x,y\in \mathbb{R}^d$, if a patch $P$ occurs in $\mathcal{U}$ while $x+P$ also occurs in $\mathcal{U}$ and $B\_r(y)\subseteq \text{supp}(P)$, then $\Vert x\Vert \leq \frac{r}{N}$ implies that $x\in \mathcal{K}$.
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