Spectral analysis of a family of binary inflation rules

Abstract

The family of primitive binary substitutions defined by $1 \mapsto 0 \mapsto 0 1^m$ with $m\in \mathbb {N}$ is investigated. The spectral type of the corresponding diffraction measure is analysed for its geometric realisation with prototiles (intervals) of natural length. Apart from the well-known Fibonacci inflation ($m=1$), the inflation rules either have integer inflation factors, but non-constant length, or are of non-Pisot type. We show that all of them have singular diffraction, either of pure point type or essentially singular continuous.

Appendix

Here, we consider some logarithmic Mahler measures, in particular $\mathfrak {m}(q)$ for the polynomial q from Eq. (5.3) with $m\in \mathbb {N}$. The polynomials q seem to be irreducible over $\mathbb {Z}$, though we have no general proof for this observation. As follows from a simple calculation, $\mathfrak {m}(q)$ takes the values $\log (3)$ for $m=1$ and $\log \bigl ( 2 + \sqrt{3}\, \bigr )$ for $m=2$. It is known that one must have $\mathfrak {m}(q) = \log (\xi )$ where $\xi$ is a Perron number. A little experimentation shows that $\xi$ is a Salem number for $m=3$, namely the largest root of $z^4 - 3 z^3 - 4 z^2 - 3 z + 1$, and a Pisot number for $m=4$, this time the largest root of $z^4 - 4 z^3 - 2 z^2 + 2 z + 1$. For $m=5$, one finds that $\xi$ is the largest root of $z^8 - 6 z^7 + 7 z^6 - 3 z^4 + 7 z^2 - 6 z + 1$, which is genuinely Perron, as the second largest root of this irreducible polynomial, with approximate value $1.354 > 1$, lies outside the unit circle. It would be interesting to know more about the numbers that show up here.

More generally, expressing $|p (t)|^2$ in Eq. (5.2) as $\bigl (\sin (m \pi t)/\sin (\pi t)\bigr )^2$, one has

$$\begin{aligned} \mathfrak {m}(q) \, = \int _{0}^{1} \log \left( 2 + \left( \frac{\sin (m \pi t)}{\sin (\pi t)} \right) ^{\! 2} \, \right) \,\mathrm {d}t . \end{aligned}$$

(5.7)

Since $\sin (m \pi t)^2 \leqslant 1$, one gets a simple upper bound as

$$\begin{aligned} \mathfrak {m}(q)\leqslant & {} \int _{0}^{1} \log \frac{1+2\, \sin (\pi t)^2}{\sin (\pi t)^2} \,\mathrm {d}t \, = \, \log (2) + \int _{0}^{1} \log \frac{2 - \cos (2 \pi t)}{1 - \cos (2 \pi t)} \,\mathrm {d}t \\= & {} \, \log (2) + \mathfrak {m}\bigl (z^2 - 4 z + 1 \bigr ) - \mathfrak {m}\bigl ( (z-1)^2 \bigr ) \, = \, \log \bigl ( 4 + 2 \sqrt{3} \, \bigr ) \, \approx \, 2.010 , \end{aligned}$$

where the logarithmic Mahler measures of the quadratic polynomials were evaluated via Jensen’s formula again. This shows that $\mathfrak {m}(q)$ is bounded for our family of polynomials. A slightly better bound can be obtained as follows.

Lemma 5.9

For any $m\in \mathbb {N}$, the logarithmic Mahler measure of the polynomial q from Eq. (5.3) satisfies the inequality $\, \mathfrak {m}(q) < \log \sqrt{46} \approx 1.914 {} 321$.

Proof

Here, we employ an argument from [10, 11] that was also used, in a similar context, in [20]. By a simple geometric series calculation, one finds that $q (z) = \frac{r (z)}{(z-1)^2}$ with

$$\begin{aligned} r (z) \, = \, z^{2m} + 2 z^{m+1} - 6 z^{m} + 2 z^{m-1} + 1 \, = \sum _{\ell =0}^{2m} c^{}_{\ell } \, z^{\ell }. \end{aligned}$$

(5.8)

Consequently, we have $\mathfrak {m}(q) = \mathfrak {m}(r) - \mathfrak {m}\bigl ( (z-1)^2\bigr ) = \mathfrak {m}(r)$.

Let $\mathfrak {M}(r) = \exp ( \mathfrak {m}(r))$ be the (ordinary) Mahler measure of r; compare [15, Sect. 1.2]. By the strict convexity of the exponential function and Jensen’s inequality, see [19, Ch. 2.2] for a suitable formulation, one finds

$$\begin{aligned} \mathfrak {M}(r) \, < \int _{0}^{1} \bigl | r (z) \bigr |_{z=\,\mathrm {e}^{2 \pi \mathrm {i}t}} \,\mathrm {d}t \, = \, \Vert r \Vert ^{}_{1} \, \leqslant \, \Vert r \Vert ^{}_{2} , \end{aligned}$$

where $r = r(t)$ is considered as a trigonometric polynomial on $\mathbb {T}$ (with the usual 1-periodic extension to $\mathbb {R}$). In fact, since r is not a monomial, we also have the strict inequality $\Vert r \Vert ^{}_{1} < \Vert r \Vert ^{}_{2}$.

Assume that $m\geqslant 2$, so that the exponents of r(z) in Eq. (5.8) are distinct. Consequently, by Parseval’s equation, we may conclude that

$$\begin{aligned} \Vert r \Vert ^{2}_{2} \, = \sum _{\ell =0}^{2m} |c^{}_{\ell }|^2 \, = \, 46 , \end{aligned}$$

so that $\mathfrak {M}(r) < \sqrt{46}$, independently of m. This inequality trivially also holds for $m=1$, and we get $\mathfrak {m}(q) = \mathfrak {m}(r) < \log \sqrt{46}$ for all $m\in \mathbb {N}$ as claimed. $\square$

With this bound, one has $\log (\lambda ) > \mathfrak {m}(q)$ for all $m\geqslant 40$, where $\lambda = \lambda ^{+}_{m}$ as before.

Remark 5.10

An even better bound can be obtained from Eq. (5.7) by observing that, as t varies a little, $\sin (m \pi t)^2$ oscillates quickly when m is large (with mean $\frac{1}{2}$), while $\bigl (\sin (\pi t)\bigr )^2$ remains roughly constant. Under the integral, one can then replace $\bigl (\sin (m \pi t)/\sin (\pi t)\bigr )^2$ by $\frac{1}{2} \bigl (\sin (\pi t)\bigr )^{-2}$, which still gives an upper bound for $\mathfrak {m}(q)$ because $\frac{\,\mathrm {d}^2}{\,\mathrm {d}t^2} \log (t) < 0$ on $\mathbb {R}_{+}$. Now,

$$\begin{aligned} \mathfrak {m}(q)\leqslant & {} \int _{0}^{1} \log \frac{3 - 2 \cos (2 \pi t)}{1 - \cos (2 \pi t)} \,\mathrm {d}t \, = \, \mathfrak {m}\bigl (z^2 - 3 z + 1 \bigr ) + \log (2) \, \\= & {} \, \log \bigl ( 3 + \sqrt{5}\, \bigr ) \, \approx \, 1.655 {} 571 , \end{aligned}$$

which is smaller than $\log (\lambda )$, where $\lambda = \lambda ^{+}_{m}$ as above, for all $m \geqslant 23$. $\diamond$

The values $\mathfrak {m}(q)$, as a function of $m\in \mathbb {N}$, seem to be increasing, so that $\lim _{m\rightarrow \infty } \mathfrak {m}(q)$ would be the optimal upper bound. The limit exists because we have $\mathfrak {m}(q) = \mathfrak {m}(r)$, and the polynomial r satisfies $r (z) = \tilde{r} (z, z^m)$ with

$$\begin{aligned} \tilde{r} (z,w) \, = \, -w \left( 6 - 2 \bigl ( z + z^{-1} \bigr ) - \bigl ( w + w^{-1} \bigr ) \right) . \end{aligned}$$

By a classic approximation theorem for two-dimensional Mahler measures, compare [15, Thm. 3.21], one has $\lim _{m\rightarrow \infty } \mathfrak {m}\bigl ( \tilde{r} (z, z^m) \bigr ) = \mathfrak {m}\bigl ( \tilde{r} (z,w)\bigr )$, where

$$\begin{aligned} \mathfrak {m}(\tilde{r} )= & {} \int _{\mathbb {T}^2} \log \bigl ( 6 - 2 \cos (2 \pi t_1) - 4 \cos (2 \pi t_2) \bigr ) \,\mathrm {d}t_1 \,\mathrm {d}t_2 \\= & {} \, 2 \int _{0}^{1} \mathrm {arsinh} \bigl ( \sqrt{2} \, \sin (\pi t_2) \bigr ) \,\mathrm {d}t_2 \, \approx \, 1.550 {} 675 . \end{aligned}$$

So, when $\mathfrak {m}(q)$ is an increasing function (which we did not prove), we immediately get the estimate $\log (\lambda ) > \mathfrak {m}(q)$ for all $m\geqslant 18$.

Remark 5.11

The polynomial s from Sect. 5.3 can be analysed in a completely analogous way. Here, one has $s (z) = - z^{\ell +1} \left( 6 - \bigl ( z + z^{-1} \bigr ) - \bigl ( w + w^{-1}\bigr ) - \bigl ( zw + (zw)^{-1} \bigr ) \right)$, and the approximation theorem results in

Also, various other properties are similar to those of the polynomial q from above. $\diamond$