2.1 Mahler measure

The topological entropy $h_{\text {top}}(\alpha _f)$ of a principal algebraic action $(X_f,\alpha _f)$ , $f\in R_d\smallsetminus \{0\}$ , coincides with its measure-theoretic entropy $h_{\unicode{x3bb} _f}(\alpha _f)$ and is given by the (logarithmic) Mahler measure of its defining polynomial f:

(2.6)

For polynomials in a single variable (that is, for $f\in R_1$ ), Mahler measure can be computed using Jensen’s formula: let $f(z)=f_0+f_1z +\cdots +f_k z^k$ with $f_0f_k\ne 0$ and (complex) roots $\unicode{x3bb} _1,\ldots , \unicode{x3bb} _k$ . Then

(2.7) $$ \begin{align} \mathsf{m}(f)=\log|f_k| + \sum_{j:\ |\unicode{x3bb}_j|>1} \log|\unicode{x3bb}_j| \end{align} $$

(cf. [Reference Lind, Schmidt and Ward21, p. 597], [Reference Schmidt29, equation (16.2)], or [Reference Young33]). The Kronecker lemma [Reference Kronecker17] states that if a polynomial $f\in R_1$ is irreducible, monic, and all its roots have absolute value at most $1$ , then f is cyclotomic, that is, for some integer n,

(2.8) $$ \begin{align} f(z)=\Phi_n(z) := \prod_{\substack{1\le \ell\le n \\ \textsf{gcd}(\ell,n)=1}} (z-e^{2 \pi i\ell/n}). \end{align} $$

Since Mahler measure is additive in the sense that $\mathsf {m}(f\cdot g)=\mathsf {m}(f)+\mathsf {m}(g)$ for all ${f,g\in R_d\smallsetminus \{0\}}$ , any $f\in R_1\smallsetminus \{0\}$ with $\mathsf {m}(f)=0$ must be a product of cyclotomic polynomials.

A similar statement is true for multivariate polynomials as well: if $f\in R_d\smallsetminus \{0\}$ , then $\mathsf {m}(f)=0$ if and only if f is a product of so-called generalized cyclotomic polynomials

$$ \begin{align*} f({\boldsymbol z})=\pm {{\boldsymbol z}^{{\boldsymbol n}_0}}\Phi_{m_1}({\boldsymbol z}^{{\boldsymbol n}_1} ) \cdots \Phi_{m_r}({\boldsymbol z}^{{\boldsymbol n}_r} ) \end{align*} $$

for some integers $m_1,\ldots , m_r$ and ${\boldsymbol n}_0,{\boldsymbol n}_1,\ldots , {\boldsymbol n}_r\in {\mathbb Z}^d$ ([Reference Boyd2], [Reference Schmidt29, Theorem 19.5], [Reference Smyth30]).

We recall the following properties of cyclic algebraic $\mathbb {Z}^d$ -action $(X_I,\alpha _I)$ (cf. [Reference Schmidt29, Ch. 6]).

• • The normalized Haar measure $\unicode{x3bb} _{X_I}$ of $X_I$ is shift-invariant.

• • If $I\subset R_d$ is non-zero and principal, $I=(f)$ , say, the topological entropy of $(X_f,\alpha _f)$ is given by the Mahler measure in equation (2.6) of f; if $I\subset R_d$ contains at least two non-zero elements $f,g$ which are relatively prime to each other (that is, without a non-trivial common factor), then $h_{\text {top}}(X_I,\alpha _I)=0$ .

• • If $d>1$ , every principal $\mathbb {Z}^d$ -action $(X_f,\alpha _f)$ is ergodic (with respect to $\unicode{x3bb} _f$ ); if $d=1$ , a principal $\mathbb {Z}$ -action $(X_f,\alpha _f)$ is ergodic if and only if f has no cyclotomic divisor.

• • For every non-zero and irreducible $f\in R_d$ , the following conditions are equivalent:

  1. – $\unicode{x3bb} _f$ is mixing under $(X_f,\alpha _f)$ ;

  2. – $h_{\text {top}}(X_f,\alpha _f)>0$ .

2.2 Main results

Our main results are the following theorems which will be proved in §§6 and 7.

For the higher dimensional case, we need an extra condition.

Definition 2.2. [Reference Lind, Schmidt and Verbitskiy20]

A non-zero Laurent polynomial $f\in R_d$ is atoral if it is not a unit in $R_d$ and its unitary variety

$$ \begin{align*} \mathsf U(f)=\{(t_1,\ldots ,t_d)\in \mathbb{T}^d\mid f(e^{2\pi it_1},\ldots ,e^{2\pi it_d})=0\} \end{align*} $$

of f has dimension $\le d-2$ . This includes the possibility that $\mathsf U(f)=\varnothing $ , which is equivalent to expansivity of the $\mathbb {Z}^d$ -action $\alpha _f$ . If $\mathsf U(f)$ has dimension $d-1$ , f is called toral.

With this definition, the following is true.

We illustrate these definitions with a few examples.

Example 2.4. (Toral automorphisms)

We start with a special case: let $f=f_0+\cdots +f_kz^k \in R_1$ with $k\ge 1$ and $f_k=|f_0|=1$ . Then the principal $\mathbb {Z}$ -action $(X_f,\alpha _f)$ is conjugate to the toral automorphism $(\mathbb {T}^k,A_f)$ , where

(2.9)

is the companion matrix of f. The map $\phi \colon X_f \to \mathbb {T}^k$ , defined by

$$ \begin{align*} \phi (x)= \left( \begin{smallmatrix} x_0 \\ \vdots \\ x_{k-1} \end{smallmatrix}\right) \end{align*} $$

for every $x=(x_n)_{n\in \mathbb {Z}}$ , implements this conjugacy. We conclude that $(X_f,\alpha _f)$ and thus $(\mathbb {T}^k,A_f)$ is Bohr chaotic if and only if $\mathsf {m}(f)>0$ (cf. equation (2.6)).

Consider now an irreducible toral automorphism $T_A\colon {\mathbb T}^k\to {\mathbb T}^k$ defined by a matrix $A\in GL_k({\mathbb Z})$ . Then the characteristic polynomial $f(z)$ of A is irreducible, and the principal algebraic $\mathbb {Z}$ -action $({\mathbb T}^k,A_f) \cong (X_f,\alpha _f)$ in equation (2.9) is a finite-to-one factor of $({\mathbb T}^d,T_A)$ . Hence, if $({\mathbb T}^k, A_f)$ is Bohr chaotic (which is the case if and only if $\mathsf (m)(f)>0$ ), then $({\mathbb T}^k,T_A)$ is also Bohr chaotic (as an extension).

If a toral automorphism $T_A\colon {\mathbb T}^k\to {\mathbb T}^k$ with $A\in GL_k({\mathbb Z})$ is reducible (that is, has a proper invariant subtorus $V\subsetneq \mathbb {T}^k$ ), then the characteristic polynomial g of $T_A|_V$ will be a proper factor of f. If $g'$ is one of the irreducible factors of g (and hence of f), then the system $(\mathbb {T}^{\text {deg}(g')},T_{A_{g'}})$ will be Bohr chaotic if and only if $\mathsf {m}(g')>0$ , in which case both $(V,T_A|_V)$ and $(\mathbb {T}^d,T_A)$ will be Bohr chaotic. By varying V over the A-invariant irreducible subtori of $\mathbb {T}^k$ , we see that $(\mathbb {T}^k,T_A)$ is Bohr chaotic if and only if $h_{\text {top}}(T_A) = \mathsf {m}(f)>0$ .

3.1 Zero entropy $\mathbb {Z}^d$ -actions are not Bohr chaotic

Consider a measure-preserving $\mathbb {N}^d$ - or ${\mathbb Z}^d$ -action $\gamma $ on a Lebesgue space $(\Omega , \mu )$ , where $\Omega $ is a compact space equipped with its Borel field. We say that the measure-theoretic system $(\Omega , \mu , \gamma )$ has completely positive entropy if any non-trivial factor of $(\Omega , \mu , \gamma )$ has positive entropy. Bernoulli systems have complete positive entropy. For $d=1$ , the following result is folklore; for $d\ge 1$ , we include a proof for completeness, based on a disjointness theorem due to Glasner, Thouvenot, and Weiss [Reference Glasner, Thouvenot and Weiss12, Theorem 1].

Proposition 3.1. Suppose that $(\Omega , \mu , \gamma )$ has completely positive entropy, $\omega \in \Omega $ is a $\mu $ -generic point, and $\phi \in C(\Omega )$ is a continuous function having zero mean. Then $(\phi (\gamma ^{{\boldsymbol n}} \omega ))_{{\boldsymbol n} \in \mathbb {N}^d}$ is orthogonal to every zero entropy $\mathbb {N}^d$ - or $\mathbb {Z}^d$ -action $(X, \alpha )$ . That is to say, for every $f\in C(X)$ and every $x\in X$ , we have

(3.1) $$ \begin{align} \lim_{N\to\infty}\frac{1}{N^d}\sum_{{\boldsymbol n} \in [0, N-1]^d} \phi(\gamma ^{{\boldsymbol n}}\omega ) f(\alpha ^{\boldsymbol n} x)=0. \end{align} $$

In particular, continuous ${\mathbb N}^d$ - or ${\mathbb Z}^d$ -actions with zero topological entropy are not Bohr chaotic.

Proof. Suppose that for some f and some x, there exists a sequence $(N_j)$ tending to infinity such that

We can assume that along this sequence $(N_j)$ , the following weak limits of measures exist:

where $\delta _\omega $ and $\delta _x$ denote the point masses at the points $\omega $ and x, respectively. Clearly, the measure $\unicode{x3bb} $ is $\gamma \negthinspace \times \negthinspace \alpha $ -invariant, and the projection of $\unicode{x3bb} $ on X is equal to $\nu $ . Since $\omega $ is $\mu $ -generic, the projection of $\unicode{x3bb} $ onto $\Omega $ is equal to $\mu $ . In other words, $\unicode{x3bb} $ is a joining of $\mu $ and $\nu $ , where $\nu $ has zero entropy. Since systems of completely positive entropy are disjoint from systems of zero entropy by [Reference Glasner, Thouvenot and Weiss12, Theorem 1], we obtain that $\unicode{x3bb} =\mu \negthinspace \times \negthinspace \nu $ . Thus, by the definition of $\unicode{x3bb} $ and the hypothesis that $\mathbb {E}_\mu \phi =0$ , we get that

$$ \begin{align*} \ell = \mathbb{E}_\unicode{x3bb} (\phi \otimes f) = \mathbb{E}_\mu \phi \cdot \mathbb{E}_\nu f =0, \end{align*} $$

which is a contradiction.

Example 3.3. (Furstenberg’s example)

Let $d=2$ and let $I = (2-z_1, 3-z_2)\subset R_2$ . Then $X_I=\{x\in \mathbb {T}^{\mathbb {Z}^2}\mid \sigma ^{(1,0)}x=2x,\,\sigma ^{(0,1)}x=3x\}$ , so that $x_{k,l}=2^k3^lx_{(0,0)}$ for every $x\in X_I$ and $(k,l)\in \mathbb {Z}^2$ . Since $f^{(2)}=2-z_1$ and $f^{(3)}=3-z_2$ are irreducible and relatively prime to each other, I is a prime ideal, and hence $h_{\text {top}}(X_I,\alpha _I)=0$ [Reference Schmidt29, Proposition 17.5].

If $\gamma $ is a continuous $\mathbb {Z}^2$ -action on a compact space $\Omega $ , $\mu $ is a probability measure on $\Omega $ with completely positive entropy under $\gamma $ , $\omega \in \Omega $ is a $\mu $ -generic point, and $\phi \in C(\Omega )$ has mean zero, Proposition 3.1 shows that

$$ \begin{align*} \lim_{N\to\infty } \frac{1}{N^2} \sum_{(m,n)\in [0,N-1]^2} \phi (\gamma ^{(m,n)}\omega )h(2^m3^nt)=0 \end{align*} $$

for every $h\in C(\mathbb {T})$ and $t\in \mathbb {T}$ .

In [Reference Furstenberg11], Furstenberg’s example was defined as the $\mathbb {N}^2$ -action $\alpha $ on $X=\mathbb {T}$ given by

$$ \begin{align*} \alpha ^{(m,n)}t=2^m3^nt\ (\text{mod 1}) \end{align*} $$

for every $(m,n)\in \mathbb {N}^2$ and $t\in \mathbb {T}$ .

We set $\Omega =\mathbb {T}^{\mathbb {N}^d}$ , write the coordinates of every $\omega =(\omega _{\boldsymbol n})_{{\boldsymbol n}\in \mathbb {N}^d}\in \Omega $ in the form $\omega _{\boldsymbol n} = (\omega _{\boldsymbol n}^{(1)},\ldots ,\omega _{\boldsymbol n}^{(d)})$ , and denote by $\gamma $ the one-sided shift-action of $\mathbb {N}^d$ on $\Omega $ (cf. equation (2.1)). According to Franklin [Reference Franklin10], for Lebesgue-almost everywhere (a.e.) $(\beta _1, \ldots , \beta _d)$ with $\beta _1>1, \ldots , \beta _d>1$ , the point $\beta =(\beta _{\boldsymbol n})_{{\boldsymbol n}\in \mathbb {N}^d} \in \Omega $ with ${\beta _{\boldsymbol n} = (\beta _1^{n_1}\;(\text {mod}\,1), \ldots , \beta _d^{n_d}\;(\text {mod}\,1))}$ for every ${\boldsymbol n}\in \mathbb {N}^d$ is Lebesgue-generic for $\gamma $ on $\Omega $ . If $\phi \colon \Omega \to \mathbb {C}$ is the map defined by

$$ \begin{align*} \phi (\omega )= e^{2\pi i(\omega_{\boldsymbol{0}}^{(1)} + \cdots + \omega_{\boldsymbol{0}}^{(d)})}, \end{align*} $$

then

$$ \begin{align*} \phi(\gamma ^{{\boldsymbol n}}\beta ) =e^{2\pi i(\beta _1^{n_1}+ \cdots +\beta_d^{n_d})} \end{align*} $$

for every ${\boldsymbol n} = (n_1,\ldots ,n_d)\in \mathbb {N}^d$ . By Proposition 3.1, the sequence $(\phi (\gamma ^{{\boldsymbol n}}\beta ))_{{\boldsymbol n}\in \mathbb {N}^d}$ is almost surely orthogonal to all systems of zero entropy. Since Furstenberg’s example $(\mathbb {T},\alpha )$ described in the preceding paragraph has zero entropy, we obtain the following corollary of Proposition 3.1.

Corollary 3.4. For almost all $(\beta _1, \beta _2)$ with $\beta _1>1$ and $\beta _2>1$ ,

$$ \begin{align*} \lim_{N\to\infty}\frac{1}{N^2}\sum_{0\le m, n<N} e^{2\pi i (\beta_1^m + \beta_2^n)} f(2^m 3^n t)=0 \end{align*} $$

for every continuous function $f \in C(\mathbb {T})$ and every $t\in \mathbb {T}$ .

3.2 Möbius disjointness and principal actions

We have just shown that zero entropy ${\mathbb Z}^d$ -actions are not Bohr chaotic. In fact, for principal actions, the result can be strengthened. We recall that a topological dynamical system $(X,T)$ is Möbius disjoint if

(3.2) $$ \begin{align} \lim_{n\to\infty} \frac 1n \sum_{k=1}^n \mu (k) f(T^kx) =0\quad \text{for every } f\in C(X) \text{ and every } x\in X. \end{align} $$

Proof. Since $(X_f,\alpha _f)$ has zero entropy, that is, $\mathsf {m}(f)=0$ , the Kronecker lemma implies that f has the form

(3.3) $$ \begin{align} f(z)= \pm z^{m_0} \Phi_{n_1}(z^{m_1})\cdots \Phi_{n_k}(z^{m_k}), \end{align} $$

where $m_0\in {\mathbb Z}$ , $n_j,m_j\in {\mathbb N}$ , $j=1,\ldots ,k$ , and $\Phi _n$ is the nth cyclotomic polynomial defined in equation (2.8). One immediately concludes from equation (3.3) that

$$ \begin{align*} f(z)=a_0+a_1z+\cdots+a_{N}z^N\quad \text{with } |a_0|=|a_N|=1, \end{align*} $$

and hence $(X_f,\alpha _f)$ is topologically conjugate to the toral automorphism $({\mathbb T}^{N},T_A)$ , where $T_A:{\mathbb T}^{N}\to {\mathbb T}^{N}$ is a linear automorphism with the matrix $A=A_{f}$ —the companion matrix of f, see Example 2.4. However, toral automorphisms with zero entropy are known to be Möbius disjoint [Reference Liu and Sarnak23, Theorem 1.1]. In fact, toral automorphisms, and more generally affine maps of compact abelian groups, are the primary examples motivating Sarnak’s (still unproven) conjecture that all topological dynamical systems with zero entropy are Möbius disjoint [Reference Sarnak28].

4.1 Riesz product measures

Let X be a compact abelian group with dual group $\widehat X$ .

Definition 4.1. [Reference Hewitt and Zuckerman13]

An infinite sequence of distinct characters $\Lambda =(\gamma _n)_{n\in \mathbb {N}}=\{\gamma _0, \gamma _1,\ldots \,\}\subset \widehat X$ is said to be dissociate if for every $k\ge 1$ and every k-tuple $(n_1,n_2,\ldots , n_k)\in \mathbb {N}^k$ of distinct non-negative integers, the equality

$$ \begin{align*} \gamma _{n_1}^{\varepsilon _1}\gamma _{n_2}^{\varepsilon _2}\cdots \gamma _{n_k}^{\varepsilon _k}=1 \end{align*} $$

with $\varepsilon _j\in \{-2,-1,0,1,2\}$ for every $j=1,\ldots ,k$ , implies that

$$ \begin{align*} \gamma _{n_1}^{\varepsilon _1}=\gamma _{n_2}^{\varepsilon _2}=\cdots =\gamma _{n_k}^{\varepsilon _k}=1. \end{align*} $$

Equivalently, $\Lambda $ is dissociate if any character in $\widehat X$ can be represented in at most one way as a finite product $ \gamma _{n_1}^{\varepsilon _1}\gamma _{n_2}^{\varepsilon _2}\cdots \gamma _{n_k}^{\varepsilon _k} $ of elements of $\Lambda $ , where all $n_j$ are distinct and ${\varepsilon _j\in \{-1, 0,1\}}$ .

Using dissociate sequences of characters, Hewitt and Zuckermann [Reference Hewitt and Zuckerman13] proposed a construction of interesting probability measures—the so-called Riesz products, generalizing Riesz products on ${\mathbb T}$ constructed by Riesz [Reference Riesz27] in 1918. More precisely, denote by $\unicode{x3bb} _X$ the Haar measure on X. Suppose that:

1. (i) $\Lambda = (\gamma _n)_{n\ge 0}$ is a dissociate sequence of characters in $\widehat X$ ;

2. (ii) $a = (a_n)_{n\ge 0}$ is a sequence of complex numbers such that $ |a_n|\le 1$ for all n.

For any $N\ge 0$ , denote by $\mu _a^{(N)}$ the measure on X which is absolutely continuous with respect to $\unicode{x3bb} _X$ with density

$$ \begin{align*} \frac {d\mu _a^{(N)}}{d\unicode{x3bb} _X}(x) = \prod_{n=0}^N(1 +{\mathsf{Re\, }} a_n\gamma _n(x)). \end{align*} $$

It is not very difficult to show that the sequence of measures $(\mu _{a}^{(N)})_{N\ge 0}$ converges weakly; the limiting measure $\mu _a=\lim _N \mu _a^{(N)}$ is called the Riesz product, and we denote it as

(4.1) $$ \begin{align} \mu _a=\prod_{n=0}^{\infty} (1 +{\mathsf{Re\, }} a_n\gamma _n(x)). \end{align} $$

The Riesz product $\mu _a$ is absolutely continuous with respect to the Haar measure $\unicode{x3bb} _X$ if and only if $\sum _{n} |a_n|^2<\infty $ , and it is singular to the Haar measure $\unicode{x3bb} _X$ if and only if ${\sum _{n} |a_n|^2=\infty }$ (see [Reference Peyrière25, Reference Zygmund34]). We will omit dependence of $\mu _a$ on the sequence $\Lambda $ , since $\Lambda $ will usually be fixed.

Since

$$ \begin{align*} 1 +{\mathsf{Re\, }} a_n\gamma _n(x) =1 + \frac {a_n}2\gamma _n(x)+ \frac {\overline a_n}2\gamma _n^{-1}(x), \end{align*} $$

the Riesz product $\mu _a$ , associated to the sequences $\Lambda $ and a, can be characterized by the Fourier coefficients $ \widehat {\mu }_a(\gamma )=\int \overline {\gamma }(x)\,d\mu _a(x)$ , $\gamma \in \widehat {X}$ , as follows.

1. (a) For any finite set of distinct characters $\{\gamma _{n_1},\gamma _{n_2},\ldots , \gamma _{n_k}\}\subset \Lambda $ and any $(\varepsilon _1, \varepsilon _2, \ldots ,\varepsilon _k)\in \{-1,0,1\}^k$ ,

   (4.2) $$ \begin{align} {\widehat\mu }_a(\gamma _{n_1}^{\varepsilon _1} \gamma _{n_2}^{\varepsilon _2}\cdots \gamma _{n_k}^{\varepsilon _k}) = a_{n_1}^{(\varepsilon _1)} a_{n_2}^{(\varepsilon _2)} \cdots a_{n_k}^{(\varepsilon _k)}, \end{align} $$
   where $a_n^{(\varepsilon )}= ({a_n}/{2}), 0,$ or ${\overline a_n}/{2}$ , depending on whether $\varepsilon =1,0$ , or $-1$ .

2. (b) For any character $\gamma \in \widehat {X}$ not of the form $\gamma _{n_1}^{\varepsilon _1}\gamma _{n_2}^{\varepsilon _2}\cdots \gamma _{n_k}^{\varepsilon _k}$ with $\varepsilon _1,\varepsilon _2,\ldots ,\varepsilon _k\in \{-1,0,1\}$ as in case (a) above, one has

   (4.3) $$ \begin{align} \widehat{\mu }_a(\gamma )=0. \end{align} $$

For any two Riesz products $\mu _a$ and $\mu _b$ , it is proved in [Reference Peyrière25] that $\mu _a$ and $\mu _b$ are mutually singular if $\sum |a_n-b_n|^2=\infty $ , and mutually equivalent if $\sum |a_n-b_n|^2<\infty $ and $\sup _n|a_n|<1$ . For any Riesz product $\mu _a$ , it is proved in [Reference Fan6] that the orthogonal series $\sum c_n (\gamma _n(x)- a_n/2)$ (with $c_n\in \mathbb {C}$ ) converges $\mu _a$ -a.e. if and only if $\sum |c_n|^2<\infty $ . Such convergence results will be useful to us in the proofs of Theorems 2.1 and 2.3. Riesz products on $\mathbb {T}$ and some generalized Riesz products appear as spectral measures of some dynamical systems (see [Reference Bourgain1, Reference Ledrappier18, Reference Queffélec26]). Riesz products are tools in harmonic analysis (see [Reference Kahane14, Reference Katznelson16, Reference Zygmund34]).

4.2 The dual group $\widehat {X}_f$

Before constructing Riesz products on $X_f$ , let us describe the dual group of $X_f$ (cf. [Reference Lind and Schmidt19, Reference Schmidt29]). Every Laurent polynomial with integer coefficients

$$ \begin{align*} h({\boldsymbol z}) =\sum_{{\boldsymbol m} \in \mathbb{Z}^d} h_{\boldsymbol m} {\boldsymbol z}^{\boldsymbol m} \in R_d \end{align*} $$

defines a character $\gamma ^{(h)} \in \widehat {\mathbb {T}^{\mathbb {Z}^d}}$ , given by

where

$$ \begin{align*} \langle h,x\rangle=\sum_{{\boldsymbol m}\in{\mathbb Z}} h_{\boldsymbol m} x_{\boldsymbol m} \end{align*} $$

for every $x\in \mathbb {T}^{\mathbb {Z}^d}$ . Conversely, every character of $\mathbb {T}^{\mathbb {Z}^d}$ is of the form $\gamma =\gamma ^{(h)}$ for some $h\in R_d$ , so that we may identify $\widehat {\mathbb {T}^{\mathbb {Z}^d}}$ with $R_d$ . Note, however, that the group operation in $R_d$ is addition, whereas in $\widehat {\mathbb {T}^{\mathbb {Z}^d}}$ , it is multiplication:

$$ \begin{align*} \gamma ^{(h+h')} = \gamma ^{(h)}\gamma ^{(h')} \end{align*} $$

for all $h,h'\in R_d$ .

Since $X_f$ is a subgroup of $\mathbb {T}^{\mathbb {Z}^d}$ , every character $\gamma ^{(h)}\in \widehat {\mathbb {T}^{\mathbb {Z}^d}}$ , $h\in R_d$ , restricts to a character $\tilde {\gamma }^{(h)}\in \widehat {X}_f$ . From the definition of $X_f$ in equation (2.2), it is clear that, for any two polynomials $h,h'\in R_d$ , $\tilde {\gamma }^{(h)} = \tilde {\gamma }^{(h')}$ if and only if $h-h'$ is a multiple of f. This allows us to identify the dual group $\widehat {X}_f$ with $R_d/(f)$ , where $(f)=R_d\cdot f$ is the principal ideal in $R_d$ generated by f:

$$ \begin{align*} \widehat{X}_f = R_d/(f). \end{align*} $$

More generally, if $I\subset R_d$ is an ideal and $X_I$ is given by equation (2.5), then

$$ \begin{align*} \widehat{X}_I= R_d/I. \end{align*} $$

4.3 Lacunary polynomials

For the construction of Riesz product measures on $X_f$ , we have to take a closer look at dissociate families $\Lambda \subset \widehat {X}_f$ in the sense of Definition 4.1.

For a given principal algebraic action $(X_f,\alpha _f)$ , where f is m-good, Riesz product measures $\mu _a$ can be constructed using the countable dissociate collection of characters $\Lambda =\{ \tilde {\gamma }^{({\boldsymbol z}^{m{\boldsymbol n}})}\mid \ {\boldsymbol n}\in \mathbb {N}^d\}$ (cf. condition (C1)). Condition (C2) ensures that any shifted family of characters $\Lambda _{{\boldsymbol k}}=\{ \tilde {\gamma }^{({\boldsymbol z}^{m{\boldsymbol n}+{\boldsymbol k}})}\mid \ {\boldsymbol n}\in \mathbb {N}^d \}$ (with ${\boldsymbol k}\in [0,m-1]^d\smallsetminus \{\boldsymbol {0}\}$ being fixed) is a $\mu _a$ -orthogonal system, as a direct consequence of equation (4.2) applied with $k=2$ —a useful property which will help us control the behavior of weighted ergodic averages. As we will see, the coefficient sequence a will be chosen depending on the non-trivial weight sequence $\boldsymbol w$ .

7.1 Homoclinic points of atoral polynomials in $R_d$ and the gap theorem

For every ${t\in \mathbb {T}}$ , we set

$$ \begin{align*} \lVert t\rVert = \min_{q\in \mathbb{Z}}|t-q|. \end{align*} $$

The existence of non-zero summable homoclinic points of $(X_f,\alpha _f)$ is equivalent to atorality of the polynomial f.

For a principal algebraic $\mathbb {Z}^d$ -action $(X_f,\alpha _f)$ , the existence of summable homoclinic points has a number of important consequences (cf. [Reference Lind and Schmidt19]): it implies positivity of entropy and very strong specification properties of the action, and it guarantees the coincidence of entropy with the logarithmic growth rate of the number of periodic points of $\alpha _f$ (that is, of points in $X_f$ with finite orbits under $\alpha _f$ —cf. [Reference Lind and Schmidt19, Reference Lind, Schmidt and Verbitskiy20]). Somewhat surprisingly, it also plays a role in the gap theorem stated below, which will imply conditions (C1) and (C2) in Definition 4.2.

We remark in passing that some of these consequences of atorality also hold for toral polynomials, but with considerably harder proofs and/or weaker conclusions—cf. e.g., [Reference Dimitrov4] or [Reference Lind, Schmidt and Verbitskiy20]). However, it is not known if specification or gap properties hold in the toral case.

To state the gap theorem referred to above, we consider, for any non-empty subset ${{\mathcal S}\subset {\mathbb Z}^d}$ and any integer $H\ge 1$ , the set $\mathcal P({\mathcal S},H)\subset R_d$ of all Laurent polynomials with support in ${\mathcal S}$ and coefficients bounded in absolute value by H:

$$ \begin{align*} \mathcal P({\mathcal S},H)=\{ v\in R_d\mid \textsf{supp}(v)\subseteq {\mathcal S}\quad\text{and}\quad \|v\|_\infty \le H\}. \end{align*} $$

For every ${\boldsymbol n}=(n_1,\ldots ,n_d)\in \mathbb {Z}^d$ , we set $\|{\boldsymbol n}\|=\max \{|n_1|,\ldots ,|n_d|\}$ . Then the following is true.

Theorem 7.3. (Gap theorem)

Suppose that $g\in R_d$ is primitive, irreducible, and atoral. For every $H\ge 1$ , there exists an integer $m\ge 1$ with the following property: for every pair of sets ${\mathcal S},{\mathcal S}'\subset \mathbb {Z}^d$ with distance

and for every $v=\sum _{{\boldsymbol n}\in {\mathcal S}\cup {\mathcal S}'} v_{{\boldsymbol n}}{\boldsymbol z}^{{\boldsymbol n}} \in \mathcal {P}({\mathcal S}\cup {\mathcal S}',H)$ which is divisible by g, the restriction of v to ${\mathcal S}$ ,

(7.1) $$ \begin{align} v_{\mathcal S}=\sum_{{\boldsymbol n}\in {\mathcal S}} v_{{\boldsymbol n}}{\boldsymbol z}^{{\boldsymbol n}}, \end{align} $$

is also divisible by g.

For the proof of Theorem 7.3, we consider the algebra $\ell ^1(\mathbb {Z}^d,\mathbb {R})$ of all functions $v\colon {\boldsymbol n} \mapsto v_{\boldsymbol n}$ from $\mathbb {Z}^d$ to $\mathbb {R}$ with $\|v\|_1=\sum _{{\boldsymbol n}\in \mathbb {Z}^d}|v_{\boldsymbol n}|<\infty $ , furnished with its usual multiplication (or convolution) $(v,w)\mapsto v\cdot w$ and involution $w\mapsto w^{*}$ , given by

(7.2) $$ \begin{align} \smash[b]{(v\cdot w)_{\boldsymbol n} = \sum_{{\boldsymbol m}\in \mathbb{Z}^d}v_{{\boldsymbol m}}w_{{\boldsymbol n} - {\boldsymbol m}} = \sum_{{\boldsymbol m}\in \mathbb{Z}^d}v_{{\boldsymbol n}-{\boldsymbol m}}w_{\boldsymbol m},} \end{align} $$

and

(7.3) $$ \begin{align} w^{*}_{\boldsymbol m} = w_{-{\boldsymbol m}} \end{align} $$

for every $v,w\in \ell ^1(\mathbb {Z}^d,\mathbb {R})$ and ${\boldsymbol m},{\boldsymbol n}\in \mathbb {Z}^d$ . If we denote by $\ell ^1(\mathbb {Z}^d,\mathbb {Z})\subset \ell ^1(\mathbb {Z}^d,\mathbb {R})$ the set of all integer-valued elements of $\ell ^1(\mathbb {Z}^d,\mathbb {R})$ and identify every $h = \sum _{{\boldsymbol n}\in \mathbb {Z}^d}h_{\boldsymbol n}\boldsymbol {z}^{\boldsymbol n}\in R_d$ with the element $(h_{\boldsymbol n})_{{\boldsymbol n}\in \mathbb {Z}^d}\in \ell ^1(\mathbb {Z}^d,\mathbb {Z})$ , we obtain an embedding

$$ \begin{align*} R_d = \ell ^1(\mathbb{Z}^d,\mathbb{Z}) \subset \ell ^1(\mathbb{Z}^d,\mathbb{R}) \end{align*} $$

in which the multiplication $(h,h')\mapsto h\cdot h'$ of Laurent polynomials extends to the composition in equation (7.2) in $\ell ^1(\mathbb {Z}^d,\mathbb {R})$ . In fact, the multiplication $(v,w)\mapsto v\cdot w$ in equation (7.2) is also well defined for $w\in \ell ^1 (\mathbb {Z}^d,\mathbb {R})$ and $v\in \ell ^{\infty } (\mathbb {Z}^d,\mathbb {R})$ , the space of all bounded sequences $(v_{\boldsymbol n})_{{\boldsymbol n}\in \mathbb {Z}^d}$ in the supremum norm $\|v\|_\infty =\sup _{{\boldsymbol n}\in \mathbb {Z}^d}|v_{\boldsymbol n}|$ , and

$$ \begin{align*} \|v\cdot w\|_\infty \le\|v\|_\infty \|w\|_1 \end{align*} $$

for all $w\in \ell ^1 (\mathbb {Z}^d,\mathbb {R})$ and $v\in \ell ^{\infty } (\mathbb {Z}^d,\mathbb {R})$ .

The shift action $\bar \sigma $ of $\mathbb {Z}^d$ on $\ell ^{\infty } (\mathbb {Z}^d,\mathbb {R})$ , defined exactly as in equation (2.1) by

(7.4) $$ \begin{align} (\bar\sigma ^{\boldsymbol m} v)_{\boldsymbol n} = v_{{\boldsymbol m}+{\boldsymbol n}} \end{align} $$

for every ${\boldsymbol m}\in \mathbb {Z}^d$ and $v\in \ell ^{\infty } (\mathbb {Z}^d,\mathbb {R})$ , extends to an action $w\mapsto w(\bar \sigma )$ of $\ell ^1(\mathbb {Z}^d,\mathbb {R})$ on $\ell ^{\infty } (\mathbb {Z}^d,\mathbb {R})$ by bounded linear operators with

$$ \begin{align*} w(\bar\sigma )= \sum_{{\boldsymbol m}\in \mathbb{Z}^d}w_{\boldsymbol m}\bar\sigma ^{\boldsymbol m}\colon \ell^{\infty} (\mathbb{Z}^d,\mathbb{R})\to \ell^{\infty} (\mathbb{Z}^d,\mathbb{R}) \end{align*} $$

for every $w\in \ell ^1(\mathbb {Z}^d,\mathbb {R})$ . Equation (7.4) implies that

$$ \begin{align*} (w(\bar\sigma )v)_{\boldsymbol n}=\sum_{{\boldsymbol m}\in \mathbb{Z}^d}w_{\textbf{m}}(\bar\sigma ^{\boldsymbol m} v)_{\boldsymbol n} = \sum_{{\boldsymbol m}\in \mathbb{Z}^d} w_{\boldsymbol m} v_{{\boldsymbol m}+{\boldsymbol n}} = (w^{*}\cdot v)_{\boldsymbol n}, \end{align*} $$

so that

(7.5) $$ \begin{align} w(\bar\sigma )v = w^{*}\cdot v \end{align} $$

for every $w\in \ell ^1(\mathbb {Z}^d,\mathbb {R})$ and $v\in \ell ^{\infty } (\mathbb {Z}^d,\mathbb {R})$ (cf. equation (7.3)).

We define a surjective group homomorphism $\eta \colon \ell ^{\infty } (\mathbb {Z}^d,\mathbb {R}) \to \mathbb {T}^{\mathbb {Z}^d}$ by setting

(7.6) $$ \begin{align} \eta (v)_{\boldsymbol n} = v_{\boldsymbol n} \;(\text{mod}\;1) \end{align} $$

for every $v=(v_{\boldsymbol n})_{{\boldsymbol n}\in \mathbb {Z}^d}$ and ${\boldsymbol n}\in \mathbb {Z}^d$ . Note that $\eta $ is shift-equivariant in the sense that

$$ \begin{align*} \eta \circ \bar\sigma ^{\boldsymbol n} = \sigma ^{\boldsymbol n} \circ \eta \end{align*} $$

for every ${\boldsymbol n}\in \mathbb {Z}^d$ ; more generally, if $w\in R_d=\ell ^1(\mathbb {Z}^d,\mathbb {Z})$ , then

(7.7) $$ \begin{align} \eta \circ w(\bar\sigma ) = w(\sigma )\eta. \end{align} $$

For every $x\in \mathbb {T}^{\mathbb {Z}^d}$ , there exists a unique point $x^{\#} \in (-\tfrac 12,\tfrac 12]^{\mathbb {Z}^d}\subset \ell ^{\infty } (\mathbb {Z}^d,\mathbb {R})$ , called the lift of x, such that

(7.8) $$ \begin{align} \eta (x^{\#}) = x. \end{align} $$

Let $g=\sum _{{\boldsymbol n}\in \mathbb {Z}^d} g_{\boldsymbol n} \boldsymbol {z}^{\boldsymbol n} \in R_d$ be the Laurent polynomial appearing in the statement of Theorem 7.3 and set

$$ \begin{align*} f=g^{*} = \sum_{{\boldsymbol n}\in \mathbb{Z}^d} g_{\boldsymbol n} \boldsymbol{z}^{-{\boldsymbol n} }. \end{align*} $$

Since $U(f)= U(g)$ , f is again atoral and has non-trivial summable homoclinic points by Theorem 7.2.

Proof. (1) Suppose that $x\in {\mathbb T}^{{\mathbb Z}^d}$ . By equation (7.7), we have

$$ \begin{align*} \eta (f(\bar{\sigma })x^{\#})=f(\sigma )\eta (x^{\#})=f(\sigma )x. \end{align*} $$

So, $x\in X_f$ , that is, $f(\sigma )x=0$ if and only if $f(\bar {\sigma })x^{\#} \in \ell ^{\infty } (\mathbb {Z}^d,\mathbb {Z})$ .

(2) If x is a non-trivial summable homoclinic point of $\alpha _f$ , then $x^\# \in \ell ^1(\mathbb{Z}^d,\mathbb{R})$ , and part (1) of this proof implies that $h=x^\#\cdot f^*\in \ell^1(\mathbb{Z}^d,\mathbb{Z}) = R_d$ . If h were divisible by $f^*$ , that is, if $x^\#\cdot f^* = h\cdot f^*$ for some $h\in R_d$ , then $(x^\#-h)\cdot f^*=0$ and [Reference Linnell and Puls22, Theorem 2.1] would imply that $x^\#=h$ and $x=\eta (x^\#)=\eta (h)=0$ . This violates our conditions on x. The converse is obvious.

Proof of Theorem 7.3

Since $f=g^{*}$ is atoral, there exists a non-trivial summable homoclinic point $x\in X_f$ . Let $x^{\#}\in \ell ^1(\mathbb {Z}^d,\mathbb {R})$ be the lift of x (cf. equation (7.8)), and let $h=f(\bar {\sigma })x^{\#}=x^{\#}\cdot f^{*} \in R_d$ (cf. Lemma 7.4). Since $x^{\#}\in \ell ^1(\mathbb {Z}^d,\mathbb {R})$ , there exists an integer $R=R(x,f,H)$ such that

$$ \begin{align*} \smash[t]{\sum_{\|{\boldsymbol n}\|\ge R} |x^{\#}_{{\boldsymbol n}}| <\frac 1{2H\|f\|_1}.} \end{align*} $$

For every non-empty subset ${\mathcal S}\subset \mathbb {Z}^d$ , we set

$$ \begin{align*} B_R({\mathcal S})=\{{\boldsymbol n}\in \mathbb{Z}^d\mid d({\boldsymbol n},{\mathcal S})=\min_{{\boldsymbol n}'\in {\mathcal S}}\|{\boldsymbol n}-{\boldsymbol n}'\|\le R\}. \end{align*} $$

Let ${\mathcal S},{\mathcal S}'\subset \mathbb {Z}^d$ be two subsets of $\mathbb {Z}^d$ with distance $d({\mathcal S},{\mathcal S}')\ge 3R$ . Suppose that a Laurent polynomial $v\in \mathcal P({\mathcal S}\cup {\mathcal S}', H)$ is divisible by $f^{*}$ , that is, that $v=\phi \cdot f^{*}$ for some $\phi \in R_d$ . Then:

1. (i) $v\cdot x^{\#}\in R_d$ ;

2. (ii) $\textsf {supp}(v\cdot x^{\#})\subset B_R({\mathcal S})\cup B_R({\mathcal S}')$ .

Indeed, item (i) follows from Lemma 7.4(2):

$$ \begin{align*} v\cdot x^{\#} = (\phi \cdot f^{*}) \cdot x^{\#}= \phi\cdot (f^{*}\cdot x^{\#})=\phi\cdot h\in R_d, \end{align*} $$

because both $\phi $ and h belong to $R_d$ . For item (ii), we note that every ${\boldsymbol n} \notin B_R({\mathcal S})\cup B_R({\mathcal S}')$ satisfies that $d({\boldsymbol n}, {\mathcal S}\cup {\mathcal S}')> R$ . Then $v_{{\boldsymbol n}-{\boldsymbol m}}=0$ for all ${\boldsymbol m}$ with $\|{\boldsymbol m}\|\le R$ , and hence

$$ \begin{align*} |(v\cdot x^{\#})_{{\boldsymbol n}}|=\bigg|\!\sum_{{\boldsymbol m}\in{\mathbb Z}^d} x^{\#}_{{\boldsymbol m}}v_{{\boldsymbol n}-{\boldsymbol m}}\bigg|\le \|v\|_\infty \sum_{\|{\boldsymbol m}\|>R}| x^{\#}_{{\boldsymbol m}}| < H\cdot \frac {1}{2H\|f\|_1}\le \frac 12. \end{align*} $$

Since $(v\cdot x^{\#})_{{\boldsymbol n}} \in \mathbb {Z}$ by item (i), it follows that $(v\cdot x^{\#})_{{\boldsymbol n}}=0$ .

Let $\psi $ be the restriction of $v \cdot x^{\#}$ to $B_R({\mathcal S})$ , and let $v_{\mathcal S}$ and $v_{{\mathcal S}'}$ be the restrictions of v to ${\mathcal S}$ and ${\mathcal S}'$ , respectively. Then $\psi \in R_d$ by item (i), and we claim that

(7.9) $$ \begin{align} \|\psi - v_{\mathcal S}\cdot x^{\#}\|_{\infty} < \frac{1}{2\|f\|_1}, \end{align} $$

that is, that

(7.10) $$ \begin{align} |\psi_{\boldsymbol n}- (v_{\mathcal S}\cdot x^{\#})_{\boldsymbol n}| < \frac{1}{2\|f\|_1} \quad \text{for every}\enspace {\boldsymbol n} \in {\mathbb Z}^d. \end{align} $$

Indeed, if ${\boldsymbol n} \in B_R({\mathcal S})$ , then $d({\boldsymbol n},B_R({\mathcal S}')) \ge R$ , and hence

(7.11) $$ \begin{align} |(v_{{\mathcal S}'}\cdot x^{\#})_{\boldsymbol n}| = \bigg|\!\sum_{{\boldsymbol m}\in {\mathcal S}'} v_{\boldsymbol m} x^{\#}_{{\boldsymbol n} -{\boldsymbol m}}\bigg|\le \|v\|_\infty \sum_{\|{\boldsymbol \ell}\|\ge R} |x^{\#}_{{\boldsymbol \ell}}|\le H\cdot \frac{1}{2 H \|f\|_1} = \frac{1}{2\|f\|_1}. \end{align} $$

Since $\psi _{\boldsymbol n} = (v_{{\mathcal S}}\cdot x^{\#})_{\boldsymbol n} + (v_{{\mathcal S}'}\cdot x^{\#})_{\boldsymbol n}$ , it follows that

$$ \begin{align*} |\psi _{\boldsymbol n} - (v_{{\mathcal S}}\cdot x^{\#})_{\boldsymbol n}| = |(v_{{\mathcal S}'}\cdot x^{\#})_{\boldsymbol n}| < \frac{1}{2\|f\|_1} \end{align*} $$

by equation (7.11). However, if ${\boldsymbol n}\notin B_R({\mathcal S})$ , then

$$ \begin{align*} |\psi _{\boldsymbol n} - (v_{\mathcal S} \cdot x^{\#})_{\boldsymbol n} |=|(v_{\mathcal S} \cdot x^{\#})_{\boldsymbol n} | = \bigg|\!\sum_{{\boldsymbol m}\in {\mathcal S}}(v_{\boldsymbol m} x^{\#}_{{\boldsymbol n}-{\boldsymbol m}})\bigg| \le H\cdot \frac{1}{2 H \|f\|_1} = \frac{1}{2\|f\|_1}. \end{align*} $$

This proves equation (7.10) for every ${\boldsymbol n}\in \mathbb {Z}^d$ .

Since both $v_{\mathcal S}\cdot x^{\#}\cdot f^{*} = v_{\mathcal S}\cdot h$ and $\psi $ lie in $R_d$ , we have that $(\psi -v_{\mathcal S}\cdot x^{\#})\cdot f^{*} \in R_d$ , but the smallness of the coordinates of $\psi -v_{\mathcal S}\cdot x^{\#}$ in equation (7.10) implies that $(\psi -v_{\mathcal S}\cdot x^{\#})\cdot f^{*}=0$ . Thus, we have proved that $\psi \cdot f^{*}=v_{\mathcal S}\cdot x^{\#}\cdot f^{*}=v_{\mathcal S}\cdot h$ , where h is not divisible by $f^{*}$ (cf. Lemma 7.4(2)). As $g=f^{*}$ is irreducible, we have proved that $v_{\mathcal S}$ is divisible by g, as claimed in the statement of this theorem.

This completes the proof of Theorem 7.3 with $m\ge 3R$ .

7.2 The conditions (C1) and (C2): divisibility by f of lacunary polynomials

According to Theorem 5.1, to prove Theorem 2.3, it suffices to prove that any irreducible and atoral polynomial $f\in R_d$ is m-good for a sufficiently large $m\in {\mathbb N}$ . Now we are going to prove this and finish the proof of Theorem 2.3.

Theorem 7.3 has an immediate corollary which implies that any atoral polynomial is m-good for sufficiently large m.

Corollary 7.5. Suppose that $f\in R_d$ is irreducible and atoral, and that $|\mathrm {supp}(f)|> 1$ . Then there exists, for every $H\ge 1$ , an integer $m\ge 1$ with the following property: for any set ${\mathcal S}\in {\mathbb Z}^d$ which is m-separated in the sense that

$$ \begin{align*} \|{\boldsymbol k}-{\boldsymbol n}\|\ge m\quad \text{for any pair } {\boldsymbol k}, {\boldsymbol n}\in{\mathcal S},\ {\boldsymbol k}\ne{\boldsymbol n}, \end{align*} $$

no non-zero polynomial $g\in \mathcal P({\mathcal S},H)$ is divisible by f.

Proof. For $H\ge 1$ and f fixed, choose m as in the statement of Theorem 7.3 (that is, $m\ge 3R$ in the proof of that theorem). Consider an arbitrary m-separated set ${\mathcal S}$ and any non-trivial polynomial $v = \sum _{{\boldsymbol n}\in {\mathcal S}}v_{\boldsymbol n}\boldsymbol {z}^{\boldsymbol n}\in \mathcal P({\mathcal S},H)$ .

If $|\textsf {supp}(v)|=1$ , then v cannot be divisible by f, since $|\textsf {supp}(f)|>1$ by assumption. Assume therefore that $|\textsf {supp}(v)|\ge 2$ , and that v is divisible by f. Since for any ${\boldsymbol n}\in \textsf {supp}(v)$ , the sets

$$ \begin{align*} \mathcal{T}=\{{\boldsymbol n}\},\quad \mathcal{T}'=\textsf{supp}(v)\smallsetminus \{{\boldsymbol n}\} \end{align*} $$

have distance at least m and hence, by Theorem 7.3, the restriction of v to $\mathcal {T}$ , that is, $v_{\mathcal {T}}=v_{{\boldsymbol n}}{\boldsymbol z}^{{\boldsymbol n}}$ must be divisible by f, which is impossible, v is not divisible by f.

The condition that $|\textsf {supp}(f)|>1$ in Corollary 7.5 is obviously necessary: the polynomial $f=2$ is obviously irreducible and atoral, and divides $2g$ for every $g\in R_d$ (irrespective of whether g is m-separated or not).

Proof. Put $H=1$ and let $m\ge 6R$ , where R is the number appearing in the proof of Theorem 7.3. Suppose $v\in \mathcal {P}(m\mathbb {Z}^d \cup (m\mathbb {Z}^d+\boldsymbol {k}),1)$ is a non-trivial polynomial divisible by f. Consider the decomposition $\textsf {supp}(v)={\mathcal S}_0\sqcup {\mathcal S}_1$ where

$$ \begin{align*} {\mathcal S}_0=\textsf{supp}(v)\cap m{\mathbb Z}^d,\quad {\mathcal S}_1=\textsf{supp}(v)\cap (m{\mathbb Z}^d+{\boldsymbol k}). \end{align*} $$

Both sets ${\mathcal S}_0,{\mathcal S}_1$ are m-separated, as subsets of $m{\mathbb Z}^d$ and $m{\mathbb Z}^d+{\boldsymbol k}$ , respectively.

We claim that for any ${\boldsymbol n}\in {\mathcal S}_0$ , there exists ${\boldsymbol n}'={\boldsymbol n}'({\boldsymbol n})\in {\mathcal S}_1$ such that $d({\boldsymbol n},{\boldsymbol n}')<3R$ . Otherwise, there exists ${\boldsymbol n}\in {\mathcal S}_0$ such that $d({\boldsymbol n}, {\mathcal S}_1)\ge 3R$ so that $d({\boldsymbol n}, \textsf {supp}(v)\smallsetminus \{{\boldsymbol n}\})\ge 3R$ . Then, by Theorem 7.3, the restriction of v to $\{{\boldsymbol n}\}$ , that is, $\pm {\boldsymbol z}^{{\boldsymbol n}}$ , is divisible by f, which is impossible. Similarly, for any ${\boldsymbol n}'\in {\mathcal S}_1$ , there exists ${\boldsymbol n}\in S_0$ such that $d({\boldsymbol n},{\boldsymbol n}')<3R$ . Thus, the support of v is a union of distinct pairs:

$$ \begin{align*} \textsf{supp}(v)=\bigcup_{{\boldsymbol n}\in{\mathcal S}_0}\{{\boldsymbol n},{\boldsymbol n}'\}, \end{align*} $$

where the distance within each pair is at most $3R$ .

Given a pair $\{{\boldsymbol n},{\boldsymbol n}'\}$ , consider the decomposition of $\textsf {supp}(v)$ :

$$ \begin{align*} {\mathcal S}=\{{\boldsymbol n},{\boldsymbol n}'\}, \quad {\mathcal S}'=\textsf{supp}(v)\smallsetminus {\mathcal S}. \end{align*} $$

The fact that $m\ge 6R$ implies $d({\mathcal S}, {\mathcal S}')\ge 3R$ . Indeed, $d({\boldsymbol n}, {\mathcal S}')= d({\boldsymbol n}, {\boldsymbol n}^{*})$ for some ${{\boldsymbol n}^{*}\in {\mathcal S}'}$ and

$$ \begin{align*} \begin{cases} d({\boldsymbol n}, {\boldsymbol n}^{*}) \ge m &\mbox{if}\ {\boldsymbol n}^{*}\in {\mathcal S}_0; \\ d({\boldsymbol n}, {\boldsymbol n}^{*})\ge d({\boldsymbol n}', {\boldsymbol n}^{*}) - d({\boldsymbol n}', {\boldsymbol n}) \ge m -3R &\mbox{if}\ {\boldsymbol n}^{*}\in {\mathcal S}_1. \end{cases} \end{align*} $$

It follows that $d({\boldsymbol n}, {\mathcal S}')>3R$ . Similarly, $d({\boldsymbol n}', {\mathcal S}')>3R$ .

Applying Theorem 7.3 to ${\mathcal S}$ and ${\mathcal S}'$ , we conclude that the restriction of v to ${\mathcal S}=\{{\boldsymbol n},{\boldsymbol n}'\}$ , that is,

$$ \begin{align*} v_{{\mathcal S}} =v_{{\boldsymbol n}} {\boldsymbol z}^{{\boldsymbol n}}+v_{{\boldsymbol n}'} {\boldsymbol z}^{{\boldsymbol n}'}, \quad v_{{\boldsymbol n}}, v_{{\boldsymbol n}'}\in\{-1,1\}, \end{align*} $$

must be divisible by f, which is impossible, since $v_{\mathcal S}$ is of the form

$$ \begin{align*} \pm {\boldsymbol z}^{{\boldsymbol m}}(1\pm {\boldsymbol z}^{{\boldsymbol \ell}}),\quad {\boldsymbol m}\in{\mathbb Z}^d, {\boldsymbol \ell}\in\mathbb{N}^d, \end{align*} $$

and hence is a product of a unit ( $\pm {\boldsymbol z}^{{\boldsymbol m}}$ ) and a generalized cyclotomic polynomial ${(1\pm {\boldsymbol z}^{{\boldsymbol \ell }})}$ , and thus must have zero Mahler measure $m(v_S)=0$ . This implies that $\mathsf {m}(f) =0$ , in violation of Theorem 7.2.