2 Outline of the argument

Our argument is part deterministic and part random. It is random in the sense that we do not explicitly construct a system $(X,\mathscr {B}, \mu , T)$ and functions $F_0, F_1, F_2$ for which the correlation sequence $C_{F_0, F_1, F_2}(n)$ is not approximable by an integral combination of nilsequences, but rather we show there are too many possibilities for the correlation functions $C_{F_0, F_1, F_2}(n)$ for this to be so.

To do this, we first explicitly construct a certain infinite sequence $\mathscr {S} \subset \mathbb {N}$ whose growth is slower than exponential in the sense that

(2.1) $$ \begin{align} \lim_{N \rightarrow \infty} \frac{|\mathscr{S}[N]|}{\log N} = \infty,\end{align} $$

where $\mathscr {S}[N] := \# \{n \in \mathscr {S} : n \leqslant N\}$ .

We show that for any choice of function $\eta : \mathscr {S} \rightarrow \{1, -\tfrac {1}{3}\}$ there is a system $(X,\mathscr {B}, \mu , T)$ and functions $F_0, F_1, F_2$ such that $C_{F_0 ,F_1, F_2}(n) = \eta (n)$ for $n \in \mathscr {S}$ .

For a random choice of $\eta $ , such a function will almost surely not be approximable by an integral combination of nilsequences. We give the details of this deduction, which uses nothing about $\mathscr {S}$ other than the growth property (2.1), in Proposition 3.1.

The heart of the argument, then, is the construction of the system $(X,\mathscr {B}, \mu , T)$ and the functions $F_0, F_1, F_2$ , given $\eta : \mathscr {S} \rightarrow \{1, -\tfrac {1}{3}\}$ . This is assembled from a sequence of finitary examples, via a (well-known) variant of Furstenberg’s correspondence principle, and here the specific nature of $\mathscr {S}$ is critical.

The idea behind the construction of these finitary examples ultimately comes from coding theory, and in particular a construction of Yekhanin [Reference Yekhanin9]. We will only need the most basic form of these ideas; for instance, we can replace all the finite-field theory in Yekhanin’s work with the simple observation that the function $\psi : \mathbb {Z} \rightarrow \{-1,1\}$ defined by $\psi (0) = 1$ , $\psi (1) = \psi (2) = -1$ , and periodic mod $3$ has the property that

$$ \begin{align*} \psi(x) \psi(x+d)\psi(x+2d) = \left\{\begin{array}{@{}ll} \psi(x), & d \equiv 0\; (\operatorname{mod}\, 3), \\ 1, & d \neq 0\; (\operatorname{mod}\, 3) .\end{array} \right.\end{align*} $$

The idea of using Yekhanin’s construction to give interesting examples in the additive combinatorics of higher-order correlations first arose in the finite field setting, in a joint work of the first author and Labib [Reference Briët and Labib5]. Those ideas have inspired the present work.

3 Entropy and nilsequences

Proposition 3.1. Let $\mathscr {S}$ be an increasing sequence of natural numbers such that

(3.1) $$ \begin{align} \lim_{N \rightarrow \infty} \frac{|\mathscr{S}[N]|}{\log N} = \infty.\end{align} $$

Then there is a function $\eta : \mathscr {S} \rightarrow \{1, -\tfrac {1}{3}\}$ such that

(3.2) $$ \begin{align} \lim_{N \rightarrow \infty} \frac{1}{|\mathscr{S}[N]|}\sum_{n \in \mathscr{S}[N]} \eta(s) a(s) = 0\end{align} $$

for all nilsequences a.

Proof The space of $C^{\infty }$ -functions on $G/\Gamma $ is dense in the space of continuous functions; to approximate a continuous function by a smooth function, average with respect to a smooth kernel supported near the identity on G. It therefore suffices to verify (3.2) for $a(n) = \phi (g^n x)$ with $\phi \in C^{\infty }(G/\Gamma )$ . Now we use the fact that there is a map

$$ \begin{align*} \mbox{Complexity} : \{ \mbox{smooth nilsequences} \} \rightarrow (0,\infty)\end{align*} $$

and a function $M : (0,\infty ) \times (0,1) \rightarrow (0,\infty )$ such that the set

$$ \begin{align*} \{ a : \mbox{Complexity}(a) \leqslant C\}\end{align*} $$

can be covered by $N^{M(C,\varepsilon )}$ balls of radius $\varepsilon $ in $\ell ^{\infty }[N]$ .

Results of this type were first observed by Frantzikinakis [Reference Frantzikinakis6, Proposition 6.2], and in fact Proposition 3.1 and its proof are very closely related to [Reference Frantzikinakis6, Theorem 1.4]. A discussion which gives what we need here is in the appendix of Altman [Reference Altman1] (note that $(g^n x_0)_{n \in \mathbb {Z}}$ is a particular example of a polynomial sequence as considered by Altman).

We pick the values of $\eta (n)$ at random, choosing $\eta (n) = -\tfrac {1}{3}$ with probability $\tfrac {3}{4}$ and $\eta (n) = 1$ with probability $\tfrac {1}{4}$ , these choices being independent for different values of $n \in \mathscr {S}$ . Then $\mathbb {E} \eta (n) = 0$ . By well-known large deviation estimates (Hoeffding’s inequality), for any fixed $1$ -bounded function b and for any distinct $n_1,\ldots , n_m$ ,

(3.3) $$ \begin{align} \mathbb{P}\bigg( \bigg|\sum_{i = 1}^m \eta(n_i) b(n_i)\bigg| \geqslant t \bigg) \ll e^{-ct^2/m}, \end{align} $$

where $c> 0$ is absolute.

Let $\omega : \mathbb {N} \rightarrow (0,\infty )$ be some function tending to infinity, to be specified later.

For each N, let $E_N$ be the following event: for all $1$ -bounded nilsequences a of complexity $\leqslant \omega (N)$ ,

(3.4) $$ \begin{align} \bigg|\sum_{n \in \mathscr{S}[N] } \eta(n) a(n)\bigg| \leqslant \frac{1}{\omega(N)} | \mathscr{S}[N] |.\end{align} $$

We estimate $\mathbb {P}(E_N)$ as follows. Pick some collection $\{ a_1,\ldots , a_J\}$ , $J \leqslant N^{M(\omega (N),1/2\omega (N))}$ of functions such that, for every $1$ -bounded nilsequence a of complexity at most $\omega (N)$ , there is some $a_i$ with $\Vert a - a_i \Vert _{\ell ^{\infty }[N]} \leqslant 1/2\omega (N)$ . Note that we do not need to assume that the $a_i$ are nilsequences (though this could be arranged if desired) and they are automatically $2$ -bounded.

If we are not in $E_N$ , there is some $a_i$ such that

(3.5) $$ \begin{align} \bigg|\sum_{n \in \mathscr{S}[N] } \eta(n) a_i(n)\bigg| \geqslant \frac{1}{2 \omega(N)} | \mathscr{S}[N] |.\end{align} $$

By (3.3), the probability of (3.5) happening, for some fixed i, is bounded above by $e^{-c' |\mathscr {S}[N]|/\omega (N)^2}$ for some $c'> 0$ . Summing over i, it follows that

$$ \begin{align*} \mathbb{P}( \neg E_N) \leqslant N^{M(\omega(N), 1/2\omega(N))} e^{-c' |\mathscr{S}[N]|/\omega(N)^2}.\end{align*} $$

Choose $\omega $ (with $\omega (N) \rightarrow \infty )$ so that

$$ \begin{align*} \frac{|\mathscr{S}[N]|}{\log N}> \frac{\omega(N)^2}{c'} ( 10 + M(\omega(N), 1/2\omega(N)))\end{align*} $$

for N sufficiently large. (Here, of course, we have used the assumption on $\mathscr {S}$ .) This then means that

$$ \begin{align*} \mathbb{P}(\neg E_N) \leqslant N^{-10}\end{align*} $$

for large N. In particular, $\sum _N \mathbb {P}(\neg E_N) < \infty $ which, by Borel–Cantelli, implies that almost surely only finitely many of the $\neg E_N$ occur. In particular, there is some particular choice of $\eta $ such that (3.4) holds for all sufficiently large N. Because every nilsequence has finite complexity, this implies the result.

To conclude this section, let us quickly sketch how one could extend Proposition 3.1 to include the case where $a()$ is a bracket polynomial or a product of such (and hence not a nilsequence with a continuous automorphic function $\phi $ ). Write $\chi _{\alpha , \beta }(n) := e(\alpha n\lfloor \beta n\rfloor )$ . The key point is that the set of functions $\chi _{\alpha , \beta }(n)$ , like the set of nilsequences of fixed complexity, has polynomially bounded covering numbers in $\ell ^{\infty }[N]$ .

To see why this is so, first note that $\chi _{\alpha , \beta }$ depends only on $\alpha (\operatorname {mod}\, 1)$ , so we may assume $0 \leqslant \alpha < 1$ . Next, replacing $\beta $ by $\beta +k$ for $k \in \mathbb {Z}$ has the effect of multiplying by a quadratic phase $e(\gamma n^2)$ (where $\gamma = \alpha k$ ). However, the set of all quadratic phases $e(\gamma n^2)$ is covered by $\ll _{\varepsilon } N^2$ balls of radius $\varepsilon $ in $\ell ^{\infty }[N]$ , because we may assume $0 \leqslant \gamma < 1$ and changing $\gamma $ by ${\varepsilon }/{N^2}$ only changes $e(\gamma n^2)$ by $O(\varepsilon )$ , uniformly for $n \leqslant N$ .

It therefore suffices to show that the covering numbers of the set $\Xi := \{ \chi _{\alpha , \beta } : 0 \leqslant \alpha , \beta < 1\}$ are polynomially bounded in $\ell ^{\infty }[N]$ . Now, restricted to $n \leqslant N$ , there are only polynomially many functions $\lfloor \beta n\rfloor $ as $\beta $ ranges in $[0, 1)$ . Indeed, the map $\beta \mapsto (\lfloor \beta n\rfloor )_{n \leqslant N}$ is only discontinuous at the points where $\beta n \in \mathbb {Z}$ for some $n \leqslant N$ , of which there are no more than $N^2$ with $0 \leqslant \beta < 1$ . Thus, $\chi _{\alpha , \beta } = \chi _{\alpha , \beta '}$ , with $\beta '$ varying in a set of size $N^2$ . Changing $\alpha $ by ${\varepsilon }/{N^2}$ only changes $\chi _{\alpha , \beta }(n)$ by $O(\varepsilon )$ , uniformly for $n \leqslant N$ . Therefore, the covering number of $\Xi $ in $\ell ^{\infty }[N]$ is $\ll _{\varepsilon } N^4$ .

It follows immediately that, for fixed C, the set of functions of type $e(\sum _{i = 1}^k \alpha _i n [\beta _i n])$ , where $k \leqslant C$ , is covered by $N^{M(C,\varepsilon )}$ balls of radius $\varepsilon $ in $\ell ^{\infty }[N]$ . One could include various types of 1-step nilsequence or bracket polynomial and obtain a similar result.

Bounds on covering numbers were all we needed to know about nilsequences, and the rest of the argument goes over verbatim.

4 The heart of the construction

Define $\psi : \mathbb {Z} \rightarrow \{-1,1\}$ to be the function with $\psi (0) = 1$ , $\psi (1) = \psi (2) = -1$ , and periodic mod $3$ . The crucial property of this function we use is the following, which is easily checked:

(4.1) $$ \begin{align} \psi(x) \psi(x+d)\psi(x+2d) = \psi(x) \end{align} $$

if $d \equiv 0\; (\operatorname {mod}\, 3)$ , and $1$ if $d \neq 0\; (\operatorname {mod}\, 3)$ .

Fix, once and for all, a sequence $M_1 < M_2 < \cdots $ of positive integers such that:

1. (1) each $M_i$ is a multiple of $3$ ;

2. (2) $\lim _{k \rightarrow \infty } k^{-2} \sum _{i=1}^k \log M_i = 0$ ;

3. (3) $\prod _{i=1}^{\infty } (1 - ({3}/{M_i})) = \gamma> 0$ .

For instance, one could take $M_i = 3i^2$ .

Define

$$ \begin{align*} \Omega_{k} := \{ (x_1,x_2,\ldots) : 0 \leqslant x_i < M_i, x_{k+1} = x_{k+2} = \cdots = 0\}.\end{align*} $$

Later on we will need the technical variant

$$ \begin{align*} \tilde \Omega_{k} := \{ (x_1,x_2,\ldots) : 0 \leqslant x_i < M_i - 3, x_{k+1} = x_{k+2} = \cdots = 0\}.\end{align*} $$

Define also $\Sigma _k$ to consist of all sequences $(x_1,x_2,\ldots )$ with precisely two non-zero entries $x_a, x_b$ , both of which equal 1, and with $x_{k+1} = x_{k+2} = \cdots = 0$ . Write

$$ \begin{align*} \Omega := \bigcup_k \Omega_{k}, \quad \tilde\Omega := \bigcup_k \tilde\Omega_{k}, \quad \Sigma := \bigcup_k \Sigma_k.\end{align*} $$

We have a bijective map

$$ \begin{align*} \beta : \Omega \rightarrow \mathbb{Z}_{\geqslant 0}\end{align*} $$

defined by

$$ \begin{align*} \beta(x_1,x_2,\ldots) = x_1 + M_1 x_2 + M_1 M_2 x_3 + \cdots .\end{align*} $$

Let $\mathscr {S} = \beta (\Sigma )$ . Thus, $\mathscr {S}$ consists of the sums of two distinct elements of the sequence $\{1, M_1, M_1M_2, M_1M_2M_3,\ldots \}$ . We claim that $\mathscr {S}$ satisfies the hypothesis (3.1) of Lemma 3.1, that is, $\lim _{N \rightarrow \infty } {|\mathscr {S}[N]|}/{\log N} = \infty $ .

To see this, let k be maximal so that $M_1 \cdots M_k \leqslant N/2$ . Then $|\mathscr {S}[N]| \geqslant \binom {k}{2}$ , whilst $\log (N/2) \leqslant \sum _{i=1}^{k+1} \log M_i$ . Therefore, it is enough that

$$ \begin{align*} \lim_{k \rightarrow \infty} k^{-2}\sum_{i=1}^{k+1} \log M_i = 0,\end{align*} $$

which follows immediately from assumption 2 above.

We now apply Proposition 3.1 to get a function $\eta : \mathscr {S} \rightarrow \{1, -\tfrac {1}{3}\}$ satisfying (3.2). Define

$$ \begin{align*} \Sigma^+_{k} := \{x \in \Sigma_k : \eta(\beta(x)) = 1\}\quad \mbox{and} \quad \Sigma^-_{k} := \big\{ x \in \Sigma_k : \eta(\beta(x)) = -\tfrac{1}{3}\big\}.\end{align*} $$

Thus, $\Sigma _k = \Sigma ^-_k\cup \Sigma ^+_k$ .

We introduce one more piece of notation. If $z \in \Sigma _k$ and if $x \in \Omega _{k}$ , then we write

$$ \begin{align*} \sigma_z(x ) := \sum_{i \in [k] : z_i = 0} x_i.\end{align*} $$

Now we come to the crucial definition. Let $k \in \mathbb {N}$ . For $x \in \Omega _{k}$ define

(4.2) $$ \begin{align} f_k(\beta(x)) = \prod_{z \in \Sigma^k_-} \psi(\sigma_z(x)).\end{align} $$

Note that $\beta (\Omega _{k}) = [0,N_k-1]$ , where

(4.3) $$ \begin{align} N_k := M_1 \cdots M_k,\end{align} $$

and so $f_k$ is a well-defined function on $[0, N_k-1]$ , taking values in $\{-1,1\}$ .

Define also the technical variant

(4.4) $$ \begin{align} \tilde f_k(\beta(x)) := 1_{x \in \tilde \Omega_k} f_k(\beta(x)).\end{align} $$

Thus, $\tilde f_k$ is defined on $[0,N_k - 1]$ and takes values in $\{-1,0,1\}$ . Extend both $f_k$ and $\tilde f_k$ to functions on all of $\mathbb {Z}_{\geqslant 0}$ by defining $f_k(n) = \tilde f_k(n) = 0$ for $n \geqslant N_k$ .

The following lemma is the heart of the argument. Here, recall that $\gamma> 0$ is just a positive constant (appearing in point 3 of the list of properties satisfied by the $M_i$ ).

Lemma 4.1. For $d \in \mathbb {Z}_{\geqslant 0}$ , write

$$ \begin{align*} S_k(d) := \frac{1}{N_k} \sum_{n \in [0,N_k-1]} \tilde f_k(n) f_k(n + d) f_k(n + 2d).\end{align*} $$

Then for $d \in \mathscr {S}$ we have $\lim _{k \rightarrow \infty } S_k(d) = \gamma \eta (d)$ .

Proof Let $d \in \mathscr {S} = \beta (\Sigma )$ . For k large enough, $d \in \beta (\Sigma _k)$ , and we assume this is so in what follows.

From the definition of $\tilde f_k$ , we see that the sum over n ranges over $n = \beta (x)$ , $x \in \tilde \Omega _{k}$ . Now for n of this form and for $d = \beta (y)$ , $y \in \Sigma _k$ , we have $x+y, x+ 2y \in \Omega _{k}$ and, moreover,

$$ \begin{align*} \beta(x + y) = \beta(x) + \beta(y) = n + d,\end{align*} $$

$$ \begin{align*} \beta(x + 2y) = \beta(x) + 2 \beta(y) = n + 2d.\end{align*} $$

Note that this ‘lack of carries’ was precisely the reason for defining the set $\tilde \Omega _{k}$ . It follows that

$$ \begin{align*} S_k(d) = \mathbb{E}_{x \in \Omega_{k}} \tilde f_k(\beta(x)) f_k(\beta(x + y)) f_k(\beta(x + 2y)),\end{align*} $$

for $d = \beta (y)$ , $y \in \Sigma _k$ . Substituting the definitions (4.2), (4.4) of $f_k, \tilde f_k$ (and noting that $\sigma _z$ is linear), we see that

$$ \begin{align*} S_k(d) = \mathbb{E}_{x \in \Omega_{k}} 1_{x \in \tilde \Omega_k} \prod_{z \in \Sigma^-_k} \psi(\sigma_z(x))\psi(\sigma_z(x) + \sigma_z(y))\psi(\sigma_z(x)+ 2\sigma_z(y)).\end{align*} $$

From (4.1) it follows that

$$ \begin{align*} S_k(d) = \mathbb{E}_{x \in \Omega_{k}} 1_{x \in \tilde \Omega_k} \prod_{z \in \Sigma^-_k: \sigma_z(y) \equiv 0 (\mbox{mod}\, 3)} \psi(\sigma_z(x)).\end{align*} $$

Now both y and z here are vectors with only two non-zero entries and so $\sigma _z(y)$ takes only the values $0,1,2$ with $\sigma _z(y) = 0$ if and only if $y = z$ . Therefore,

(4.5) $$ \begin{align} S_k(d) = \bigg\{ \begin{array}{@{}ll} \mathbb{E}_{x \in \Omega_{k}} 1_{x \in \tilde\Omega_k} \psi(\sigma_y(x)) & \text{if } y \in \Sigma^-_k,\\ \mathbb{E}_{x \in \Omega_{k}} 1_{x \in \tilde\Omega_{k}} & \text{if }y \in \Sigma^+_k.\end{array} \end{align} $$

The second expression is

$$ \begin{align*} \mathbb{E}_{x \in \Omega_{k}} 1_{x \in \tilde\Omega_{k}} = \frac{|\tilde \Omega_k|}{|\Omega_k|} = \prod_{i=1}^k \bigg(1 - \frac{3}{M_i}\bigg) \rightarrow \gamma\end{align*} $$

as $k \rightarrow \infty $ . The first expression in (4.5) may be written explicitly as

(4.6) $$ \begin{align} \frac{|\tilde \Omega_k|}{|\Omega_k|}\mathbb{E}_{x \in \tilde\Omega_k} \psi(x_1 + \cdots + \hat{x}_i + \cdots + \hat{x}_j + \cdots + x_k),\end{align} $$

where y has non-zero coordinates at $i,j$ and the hat means that $\hat {x}_i$ does not appear in the sum. Note, however, that $\tilde \Omega _k$ is a box with sidelengths $M_i - 3$ , each of which is a multiple of $3$ . Therefore $x_1 + \cdots + \hat {x}_i + \cdots + \hat {x}_j + \cdots + x_k$ is uniformly distributed mod $3$ , as x ranges uniformly over $\tilde \Omega _k$ , and the average in (4.6) is

$$ \begin{align*} \frac{|\tilde \Omega_k|}{|\Omega_k|} \cdot \bigg(\!-\frac{1}{3}\bigg) = -\frac{1}{3} \prod_{i=1}^k \bigg(1 - \frac{3}{M_i}\bigg) \rightarrow -\frac{\gamma}{3}.\end{align*} $$

This completes the proof.

5 Putting everything together

Our final task is to build a measure-preserving system from the functions constructed in the last section. For this we need a slight variant of the usual Furstenberg correspondence principle, proven in a very similar way. An essentially equivalent statement may be found, for instance, in [Reference Frantzikinakis8, Proposition 3.3].

Proposition 5.1. Let $A \subset \mathbb {R}$ be a finite set. Suppose that for each $k \in \mathbb {N}$ we have functions $f_{0,k}, \cdots , f_{r,k} : \mathbb {Z}_{\geqslant 0} \rightarrow A$ , and that $(N_k)_{k=1}^{\infty }$ is an increasing sequence of positive integers. Then there is a measure-preserving system $(X, \mathscr {B}, \mu , T)$ and functions $F_0,F_1,\ldots , F_r \in L^{\infty }(\mu )$ such that the following is true: if $(d_1,\ldots , d_r)$ is a tuple of distinct positive integers such that

$$ \begin{align*} S(d_1,\ldots, d_r) := \lim_{k \rightarrow \infty} \frac{1}{N_k} \sum_{n \in [0,N_k-1]} f_{0,k}(n) f_{1,k}(n + d_1)\cdots f_{r,k}(n + d_r)\end{align*} $$

exists, then

$$ \begin{align*} S(d_1,\ldots, d_r) = \int_X F_0 \cdot T^{d_1} F_1 \cdot T^{d_2} F_2 \cdots T^{d_r} F_ r d\mu.\end{align*} $$

We apply this with the functions constructed in the last section, taking $r = 2$ , $f_{0,k} := \tilde f_k$ , $f_{1,k} = f_{2,k} = f_k$ and $N_k = M_1 \cdots M_k$ as before.

By Proposition 5.1 and Lemma 4.1, there is a measure-preserving system $(X,\mathscr {B}, \mu , T)$ together with functions $F_0, F_1, F_2 \in L^{\infty }(\mu )$ such that, writing $C_{F_0, F_1, F_2}(d) := \int _X F_0 \cdot T^d F_1 \cdot T^{2d} F_2\,d\mu $ , we have

(5.1) $$ \begin{align} C_{F_0, F_1,F_2}(d) = \eta(d) \quad \text{for } d \in \mathscr{S}.\end{align} $$

(Note it is clearly possible to scale the $F_i$ to remove $\gamma $ factor appearing in Lemma 4.1.) We claim that it is impossible to write

$$ \begin{align*} C_{F_0, F_1, F_2}(n) = a(n) + b(n)\end{align*} $$

with a an integral combination of $2$ -step nilsequences and $\Vert b \Vert _{\infty } \leqslant {1}/{100}$ . Suppose that this were possible. Then, from (5.1) and the fact that $\eta $ takes values in $\{1, -\tfrac {1}{3}\}$ , we would have $(a(d) + b(d)) \eta (d) \in \{\tfrac {1}{9}, 1\}$ for all $d \in \mathscr {S}$ . However, $|b(d) \eta (d)| \leqslant {1}/{100}$ and, therefore,

(5.2) $$ \begin{align} \Re (a(d) \eta(d) ) \geqslant \frac{1}{9} - \frac{1}{100}> \frac{1}{10}\end{align} $$

for all $d \in \mathscr {S}$ .

Suppose that

$$ \begin{align*} a(n) = \int_M a_m(n)\,d\sigma(m).\end{align*} $$

Here, M is a compact metric space, $\sigma $ is a complex Borel measure of bounded variation and the $a_m$ are nilsequences, with the map $m \mapsto a_m(n)$ being in $L^{\infty }(\sigma )$ .

Then (5.2) implies that

$$ \begin{align*} \bigg| \frac{1}{|\mathscr{S}[N]|} \sum_{n \in \mathscr{S}[N]} a(n) \eta(n) \bigg| \geqslant \frac{1}{10}.\end{align*} $$

On the other hand, we have

$$ \begin{align*} \bigg| \frac{1}{|\mathscr{S}[N]|} \sum_{n \in \mathscr{S}[N]} a(n) \eta(n) \bigg| \leqslant \int_M \bigg| \frac{1}{|\mathscr{S}[N]|} \sum_{n \in \mathscr{S}[N]} a_m(n) \eta(n) \bigg| \, d|\sigma|. \end{align*} $$

However, by the choice of $\eta $ (Lemma 3.1) we have

$$ \begin{align*} \lim_{N \rightarrow \infty} \frac{1}{|\mathscr{S}[N]|} \sum_{n \in \mathscr{S}[N]} a_m(n) \eta(n) = 0\end{align*} $$

for all m. Therefore, by the dominated convergence theorem,

$$ \begin{align*} \lim_{N \rightarrow \infty} \int_M \bigg| \frac{1}{|\mathscr{S}[N]|} \sum_{n \in \mathscr{S}[N]} a_m(n) \eta(n) \bigg| \, d|\sigma| = 0.\end{align*} $$

Putting these statements together gives a contradiction, and this completes the proof of Theorem 1.2.