Ergodic Properties of $k$-Free Integers in Number Fields

Let $K/\mathbf Q$ be a degree $d$ extension. Inside the ring of integers $\mathcal O_K$ we define the set of $k$-free integers $\mathcal F_k$ and a natural $\mathcal O_K$-action on the space of binary $\mathcal O_K$-indexed sequences, equipped with an $\mathcal O_K$-invariant probability measure associated to $\mathcal F_k$. We prove that this action is ergodic, has pure point spectrum and is isomorphic to a $\mathbf Z^d$-action on a compact abelian group. In particular, it is not weakly mixing and has zero measure-theoretical entropy. This work generalizes the paper by the first author and Sinai arXiv:1112.4691 [math.DS] where $K=\mathbf Q$ and $k=2$.


Introduction
It is an interesting question to study "randomness" of a given deterministic sequence. For a typical sequence coming from a chaotic dynamical system, such as doubling modulo one, one expects strong statistical properties, while for a circle rotation such properties cannot be expected. Of particular interest in this setting is the Möbius sequence, {µ(n)} n 1 defined as if n is not square-free; (−1) m , if n is the product of m distinct primes.
It is well known that n N µ(n) = o(N ), (1) suggesting that {µ(n)} n 1 is reminiscent of a sequence of zero mean iid random variables, the above statement being the Law of Large Numbers for such a sequence. A related sequence, {µ 2 (n)} n 1 , has been investigated by Sinai and the first author [3]. Their main result is that {µ 2 (n)} n 1 , which is a sequence of zeros and ones, is generic for an ergodic subshift of infinite type on {0, 1} Z with pure point spectrum (see Section 3.2 for more details). Such systems had been studied by von Neumann and Halmos [22,5], and their statistical properties are well understood: they have zero measure-theoretical entropy and are not weakly mixing. In other words, the sequence {µ 2 (n)} n 1 has as little "randomness" as possible.
In the present paper we generalize the main result of [3] in two directions. Firstly, realizing that µ 2 (n) is the indicator of square-free integers, we write µ (k) (n) for the indicator of k-free numbers; that is, numbers that are not divisible by p k for every prime p. Secondly, we pass to a degree d number field K/Q with ring of integers O K and define the Möbius function on O K . Since ideals a in O K factor uniquely, we can define µ by if a is not square-free; (−1) m , if a is the product of m distinct prime ideals and µ (k) by µ (k) (a) = 1, p k ⊇ a for every prime ideal p; 0, otherwise.
Then, for a ∈ O K , set µ(a) = µ((a)) and µ (k) (a) = µ (k) ((a)). An ideal a is k-free if µ (k) (a) = 1, while an integer a ∈ O K is k-free if the principal ideal (a) is k-free, and we denote the set of k-free integers in O K by F k . Thus {µ (k) (a)} a∈O K is an O K -indexed sequence of zeros and ones.
By an O K -subshift we mean a shift-invariant probability measure P on X = {0, 1} O K or, equivalently an action O K (X, P). Let ι : (Z d , +) → (O K , +) be a group isomorphism, where d is the degree of the extension K/Q; it is unique up to multiplication by an element of Aut(Z d ). The group Z d acts via ι on the space of O K -indexed sequences by d commuting translations, and every O K -subshift corresponds to a Z d -subshift. Let B x denote the ball of radius x centered at the identity with respect to the L 1 norm induced on O K after identification with Z d inside R d via ι. We say that a sequence z = {z(a)} a∈O K ∈ {0, 1} O K is generic for an ergodic O K -subshift P if the ergodic theorem holds for z, i.e. for every a 1 , . . . , a r ∈ O K , lim x→∞ 1 #B x a∈Bx z(a + a 1 ) · · · z(a + a r ) = P{w ∈ {0, 1} O K : w(a 1 ) = . . . = w(a r ) = 1}. (2) In other words, genericity means that the frequency of every finite block equals the measure of the corresponding cylinder according to the subshift. This notion does not depend on the choice of ι. A sequence z for which the limit on the LHS of (2) exists is called stationary. Given a stationary sequence z, the subshift P satisfying (2) is uniquely defined by Kolmogorov consistency [8,9] up to sets of measure zero, and in particular does not depend on ι. Our main theorem states that the sequence {µ (k) (a)} a∈O K is stationary and, more importantly, that the corresponding subshift is ergodic and has pure point spectrum.
Theorem 1.1 (Main Theorem, first version). Let K/Q be a degree d extension.
(i) There exists a unique O K -subshift Π such that the sequence {µ (k) (a)} a∈O K is generic for Π. This subshift is ergodic and has pure point spectrum.
(ii) The O K -subshift Π is isomorphic to an action of Z d by commuting translations on a compact abelian group equipped with the Haar measure.
The proof of Theorem 1.1 and of its full version Theorem 7.1 explicitly constructs the pure point subshift Π. The argument consists of three steps.
First, we show the stationarity of the sequence {µ (k) (a)} a∈O K by proving the existence of the asymptotic frequencies (correlation functions) We compute c r+1 (a 1 , . . . , a r ) explicitly (they do not depend on ι) and give an error term for finite x in Theorem 4.3. This theorem is of independent interest, along with other explicit formulae given in Section 6 (e.g. Proposition 6.1 generalizing a theorem by Hall [4]).
A particular case of (3) is c 2 (0) = 1/ζ K (k) (Corollary 4.2), stating that the density of k-free integers in O K is 1/ζ K (k), where ζ K is the Dedekind zeta function for the number field K/Q. For K = Q, the study of the average in (3) as x → ∞ is classical, see [12,4,21,6].
There is another notion of k-freeness for points in an arbitrary lattice studied by Baake, Moody, and Pleasants [1] and by Pleasants and Huck [16], for which the second correlation function, along with entropies and diffraction spectra, has been computed explicitly. This notion of k-freeness agrees with the one discussed above only when K = Q.
The next step is to construct the compact abelian group where the direct product ranges over prime ideals p in O K . We do this in Section 5 using only the second correlation function, Bochner theorem, and Pontryagin duality. Since ideals thought of as additive subgroups have finite index, each factor O K /p 2 is a finite group under addition. The Haar measure on G is simply the product of the counting measures on each factor. By identifying O K with Z d as a group, and by choosing a basis for Z d , we get an action Z d (G, Haar). By construction, the spectrum of this action is pure-point, given by the countable group Λ =Ĝ which can be identified with a subset of the d-dimensional torus.
In the third step (Section 7), we consider the unique probability measure Π on X = {0, 1} O K whose finite dimensional marginals agree with the correlation functions above: for every r 0 and every a 0 , a 1 , . . . , up to normalization. This defines a unique O K -subshift (an action O K (X, Π)) for which the d-dimensional sequence {µ (k) (a)} a∈O K is generic. A substantial part of Section 7 is dedicated to showing that the spectrum of the action O K (X, Π) is given by Λ (Theorem 7.1). The method employed is constructive and uses explicit formulae for the two and three point correlation functions. Then, we apply a theorem of Mackey's [11], which states that two actions with pure point spectrum are isomorphic if and only if they are isospectral. Since we know that Z d (G, Haar) has spectrum Λ, Theorem 1.1 follows from the isomorphism. A consequence of the Main Theorem is the in Theorem 1.1 is not weakly mixing and it has zero measure-theoretical entropy.
The corollary follows immediately: see, e.g. [2] to get absence of weak mixing and [23] to get zero measure-theoretical entropy. In the case of rational integers Corollary 1.2 was also proven by Sarnak [19]. Corollary 1.2 suggests that any randomness in the Möbius function comes from the distribution of ±1's, and not from the locations of zeros. In the context of rational integers this is expressed by a generalization of (1): Conjecture 1 (Chowla). For every n 1 , . . . , n r ∈ N and k 1 , . . . , k r ∈ {1, 2} not all even N n=1 µ k 1 (n + n 1 )µ k 2 (n + n 2 ) · · · µ kr (n + n r ) = o(N ) This conjecture, whose only proven instance is (1), implies a recent conjecture by Sarnak: [19]). Let (X, T ) be a compact topological dynamical system with zero topological entropy. Let ξ(n) = f (T n x), where x ∈ X and f ∈ C(X, C). Then the sequence {ξ(n)} n 1 does not correlate with the Möbius function, i.e. Sequences {ξ(n)} n 1 as above are called deterministic. It is known that Conjecture 2 holds true for a wide class of deterministic sequences (see, e.g., [10] and references therein).
It is worthwhile to stress the link between topological and measure-theoretical dynamics. For the case of K = Q, it is known that the topological subshift of infinite type obtained by orbit closure of {µ 2 (n)} n 1 inside {0, 1} Z has positive topological entropy 6 π 2 log 2 (see [19]) and, by the variational principle, one can find invariant probability measures with smaller measure-theoretical entropy. The measure Π defined in (4), which Sinai and the first author consider in [3] and for which {µ 2 (n)} n 1 is generic, has zero entropy and is the Pinsker factor (largest zero entropy factor) of the measure of maximal entropy. In fact, Peckner [15] showed that the measure of maximal entropy is a Bernoulli extension of Π. This means that the subshift of infinite type in Theorem 1.1 is, at least in the case K = Q, a building block for other relevant systems. It would be of interest to extend this result to arbitrary number fields K, where the strictly 1-dimensional method of [15] cannot be applied directly.
Section 2 illustrates the results in the case of square-free Gaussian integers. Some background on ideals in O K and group actions with pure point spectrum is given in Section 3, which may be skipped by readers familiar with these topics. In Section 4, we show that limits of correlations exist for the sequence {µ (k) (a)} a∈O K . In Section 5, we construct the spectral measure for the O K -shift and an abstract dynamical system having this spectral measure. Section 6 contains computations that are used in Section 7 to prove Theorem 1.1 and its more detailed version Theorem 7.1.  In this case we can write the Dedekind zeta function in terms of primes in Z. For s > 1 we have where ζ denotes the Riemann zeta function and β the Dirichlet beta function, The density of square-free Gaussian integers is 1/ζ Q(i) (2) = 6 π 2 G ≈ 0.6637, where G = β(2) is the Catalan constant.
Let us look at correlation functions. For example, c 4 (1, i, 1 + i) = 0 because for every a ∈ Z[i] at least one of the four Gaussian integers a, a + 1, a + i, a + 1 + i, is not square-free, since it is divisible by 2 = (1 + i) 2 . We will show in Proposition 4.1 that c r+1 (a 1 , . . . , a r ) = where D(p 2 | 0, a 1 , . . . , a r ) is the number of distinct residue classes among 0 + p 2 , a 1 + p 2 , . . . , a r + p 2 in Z[i]/p 2 . The fact that c 4 (1, i, 1 + i) = 0 can be derived by the formula above. In fact, notice that D(p 2 | 0, 1, i, 1 + i) = 4 for all prime ideals p and that there is a prime ideal, p 1 = (1 + i), for which N (p 2 1 ) = 4. On the other hand, there are only three distinct residue classes, whilst for every prime ideal p = p 1 we have D(p 2 | 0, 1, i, −1, −i) = 5 and N (p) 5. More precisely , Π) is simply given by the two commuting translations a + ib → (a + 1) + bi and a + bi → a + (b + 1)i, under which the probability measure Π is invariant. By construction, the 2-dimensional sequence {µ 2 (a + bi)} a+bi∈Z[i] is generic for this action. Consider now the group It is the direct product of finite abelian groups Z . . and it is acted upon coordinate-wise by Z 2 via ι. , Π) and Z 2 (G, Haar) are isomorphic. More precisely, they have pure point spectrum given by the discrete group Λ =Ĝ, identified via ι with a subset of T 2 (viewed as Z 2 ).
Here is the explicit construction of Λ in this example. For every square-free ideal d ⊆ Z[i], view d as subgroup of Z 2 and consider a fundamental domain F d 2 for Z 2 /d 2 , say the square with sides w = (w 1 , w 2 ), and w = (−w 2 , w 1 ), where w 1 > 0, w 2 0, and w 2 1 + w 2 2 = N (d). In this way #F d 2 = N (d 2 ). One can check that the annihilator of the ideal d 2 in Z[i] can be identified with a subset of Z 2 = T 2 and written as See Figure 2 for an example. The spectrum Λ is the subgroup of T 2 obtained as union of the (d 2 ) ⊥ 's as above, and is shown in Figure 3.

Ideals in Number Fields
Let K be a number field of degree [K : Q] = d, and let O K be its ring of integers. While O K need not be a principal ideal domain, it is always a Dedekind domain, that is an integral domain whose proper ideals factor (uniquely, up to the order) into a product of prime ideals (see [13,14]). We will denote ideals in O K by a, b, c, d, n, and prime ideals by p, q. The sum of two ideals is defined as n ∈ N}, and these are ideals. We say that a divides b if and only if b = ac for some ideal c or, equivalently, if and only if a ⊇ b. For any two ideals a, b we define their greatest common divisor as the smallest ideal containing both a and b, that is gcd(a, b) = a + b. The least common multiple of a and b is defined as the largest ideal contained in both a and b, We will say that two integers a, b ∈ O K are congruent modulo the ideal a (denoted by where #· denotes the cardinality of a finite set. There is also a notion of norm for elements of K. Let a ∈ K and let m a : K → K be the Q-linear map m a (b) = ab. The norm N K/Q (a) is defined as the determinant of m a . If K/Q is Galois, then N K/Q equals the product of the conjugates of a. Note that N K/Q (a) need not be positive. When O K is a principal ideal domain, and a = (a), then N (a) = N K/Q (a). The Dedekind zeta function is given, for s > 1, by where the sum ranges over all nonzero ideals of O K and the product over the prime ones. Let us also set N (a) = N ((a)).

Group Actions with Pure Point Spectrum
Let us recall the notions of ergodic and pure point spectrum actions relevant to our setting. Let G be a separable locally compact abelian group and let S be a standard Borel G-space.
. When U is a discrete direct sum of finite-dimensional irreducible representations one says that the action G (S, µ) has pure point spectrum. This means that there exists an orthonormal basis The group Γ is referred to as the spectrum of the action.
One can construct ergodic actions with pure point spectrum as follows. Let K be a compact group, H ⊆ K a closed subgroup, ϕ a continuous homomorphism of G onto a dense subgroup of K. Define the action G One can check that the latter is ergodic (by transitivity of the action) and has pure point spectrum (the unitary representation U of G on ; since V is a subrepresentation of the regular representation of K, it decomposes into a direct sum of finite-dimensional irreducible representations by the Peter-Weyl Theorem). Mackey [11] proved that, modulo removing null sets in S and K/H, every ergodic action with pure point spectrum can be realized as above. His work generalized the classical theory by von Neumann [22] and Halmos and von Neumann [5] where G = Z. In all these cases, for actions with pure point spectrum, the isomorphism class is uniquely determined by its spectrum.

Arithmetical Pattern Problems for k-Free Ideals
We need a notion of size on O K with the property that any ball of finite radius is finite. This is in general not true for the algebraic norm N , as there are number fields whose group of units is infinite. To avoid this problem, we consider a geometric norm on · by viewing O K as a vector space over Q. Let For any choice of generators we call the set ∆ = { j ε ij v j : ε ij = 0 or 1} a cell of a. The diameter of an ideal a, written diam a, is defined by min ∆ is a cell of a max a∈∆ a .
We introduce several abbreviations to simplify notation. We write a = (a 1 , . . . , a s ) ∈ (O K ) s , n = (n 1 , . . . , n s ) for s-tuples of ideals of O K , n k = (n k 1 , . . . , n k s ) for s-tuples of powers of ideals, and µ(n) = s i=1 µ(n i ) for the product Möbius function. We also tacitly set the range for integers i and j to be {1, . . . , s}. Define the function D by D(a | a) = #{b mod a | b ≡ a i mod a for some i} and more generally set D(a, b | a) = #{b mod a | b ≡ a i mod a for some i and b ∈ gcd(a, b)}.
We shall refer to c r+1 as the (r + 1)-st correlation function for the set of k-free integers in O K ; we will not indicate the dependence on k explicitly. By taking r = 1 and a 1 = 0 in Proposition 4.1, we have the well known where ζ K is as in (6).
We will actually prove a more general version of Proposition 4.1, namely a quantitative asymptotic statement on the frequencies of arbitrary binary configurations in {µ (k) (a)} a∈O K with an additional divisibility constraint (Theorem 4.3 below). Set for positive S k,b (a) computed in (16) and every ε > 0.
For ideals n 1 , . . . , n s define the E symbol by Lemma 4.4. Equation (14) evaluates to 1 precisely when a i − a j ≡ 0 mod gcd(n i , n j ) for every i and j. In this case there is exactly one b in each residue class modulo lcm(n).
Proof. It is enough to observe that is an isomorphism onto its image.
x and b + a i ≡ 0 mod n i for all i} with notation as before. Then Proof. We omit the proof as it is standard.
Lemma 4.6. Let T (x) be the number of solutions to the system such that b x. Then we have Proof. Apply Lemmata 4.4 and 4.5 with s replaced by s + 1. Proof. Follows by a simple induction.
Lemma 4.8. Take t 1 and m 2. Also fix a prime ideal p and k 1 , . . . , k t ∈ O K such that Then η 1 ,...,ηt∈{0,1} not all zero Proof. Note that η i is non-zero for at least one i, so that Lemma 4.9. With notation as before set Then we have and it vanishes precisely when for some p.
From (18) it follows that ν i = 0 for all i or ν j = 0 for all j, so the preceding equation is verified.
For the second claim (20) note that if the left-hand side vanishes, then so does the righthand side. So suppose the left-hand side doesn't vanish, that is, a i − a j ∈ gcd(n i , n j ), a i ∈ gcd(b, n i ) a i − a j ∈ gcd(n i , n j ), a i ∈ gcd(b, n i ) by Lemma 4.4. From (18) these conditions are equivalent to a i − a j ∈ gcd(n i n i , n j n j ), a i ∈ gcd(b, n i n i ), and the claim follows by another application of Lemma 4.4. Now then we write This sum converges absolutely We evaluate ψ p . Observe that the terms of (22) are zero if a j are not congruent modulo p k for all j ∈ {i : δ i = 1}. For ρ modulo p k let t ρ denote the number of integers a 1 , . . . , a s that are congruent to ρ. If t ρ > 0, denote them by k

Thus we have
It remains to verify the positivity part. Clearly the product vanishes if one of the factors does. Otherwise the factors cannot be less than 1 − s N k (p) for N (p) large enough, and these give a non-zero product.  Proof. Fix b ∈ O K whose irreducible polynomial has degree d (it exists by the Primitive Element Theorem). Let Ξ = Z[a, ab, ab 2 , . . . , ab d−1 ]. Observe that Ξ < O K is a finite index additive subgroup and that Ξ ⊆ (a), implying that #O K /Ξ N ((a)). Using Lemma 4.10 we have as needed. Note that minimizing d−1 i=0 b i over b with full degree irreducible polynomial will remove the dependence on the choice of b.
Proof of Theorem 4.3. Say D(p k , b | a) = N (p m ) N (gcd(p k ,b)) for some p. Then it is easy to see that a + a i ∈ p k for some i, proving the first case.
Suppose then that D(p k , b | a) < N (p k ) N (gcd(p k ,b)) for all p. Let α ∈ (0, 1/k) to be chosen later. From the relation (5) we get that The first sum is over s-tuples of ideals n i of norm at most x α , while Σ 2 includes s-tuples where at least one ideal has norm greater than x α . By Lemma 4.6, The first error term from (24) is at most a constant times The second error term is bounded by For Σ 2 we have Since d(n) ε N ε (n), we can bound this by ε N (n j )>x α a+a j ∈n k j a x i =j N ε ((a + a i )), and from Lemma 4.11 we have The three error terms come from equations (25)

Construction of Λ and the action Z d (G, Haar)
In this section, using only the second correlation function, we construct the groups Λ and G.
Then we discuss an action Z d (G, Haar) which has Λ as spectrum.
For an ideal a ⊆ O K , let us consider the annihilator a ⊥ , i.e. the set of unitary characters χ : O K → S 1 such that χ(a) = 1 for all a ∈ a, see [20]. Notice that #a ⊥ = #O K /a = N (a). Throughout the paper, d indicates a square-free ideal; equivalently, d can be thought as a finite collection of prime ideals (or places).
Lemma 5.1. Let us consider the measure Then ν(a) = c 2 (a), a ∈ O K .
We shall refer to ν as the spectral measure. Before proving this lemma, we need two additional results. First, it is convenient to have another formula for σ d as an Euler product. .
Proof. Multiply out the product in the RHS to get the sum (28) defining σ d .
In particular, Lemma 5.2 shows that σ d is positive and bounded away from zero and infinity. More precisely and we can also write The second correlation function is the Fourier transform of a spectral measure whose atoms are weighted by the quantities σ d . The following lemma allows us to write c 2 directly in terms of σ d .
and sum converges absolutely.
Proof. From Proposition 4.1 we get This gives .
The sum in the RHS of (30) converges absolutely by (29). By Lemma 5.2 we have where the last equality comes from (31).
Proof of Lemma 5.1. Using Lemma 5.3 we can write The function C d is constant (equal to σ d ) on the lattice d k and zero elsewhere. This function on O K is the Fourier transform of a measure on O K , given by a sum of Dirac δ-measures at the points in the set (d k ) ⊥ , with equal weights equal to σ d /N (d k ). The formula for the spectral measure ν and the lemma are proved.
Let Λ be the support of the spectral measure ν defined above. It is automatically a group and, by the Chinese Remainder Theorem for ideals, Let us remark that the union in (32) is not disjoint. It will be useful for us to single out the smallest annihilator to which a character belongs. To this extent, let us notice that if d 1 ⊇ d 2 then (d k 1 ) ⊥ ⊆ (d k 2 ) ⊥ and let us define the reduced annihilator as In other words By Pontryagin duality (see, e.g., [7]), Λ is isomorphic to the compact abelian group Elements of G are coset sequences indexed by the set of prime ideals in O K , i.e. g = (g p k +p k ) p , where g p k + p k ∈ O K /p k . Given h ∈ G, we denote by T h the translation T h (g) = g + h.
The Haar measure on G is the product of the counting measures on each factor O K /p k and is defined on the natural Borel σ-algebra on G.
We have a Z d -action on Z d (G, Haar) as follows: if v ∈ Z d and g = ( In other words, Z d acts by d commuting translations T u 1 , . . . , T u d on G, where u i = (e i + p k ) p ∈ G.
Let us now discuss the spectrum of the action (35). For v ∈ Z d let U v be the unitary operator on H = L 2 (G, Haar) given by Proposition 5.4. The spectrum of Z d (G, Haar) is isomorphic to Λ.
Proof. Let ι : Z d → O K be the isomorphism defined in the Section 4. Let g ∈ G and for every prime ideal p let g p k +p k ∈ O K /p k be its projection onto the p k -th coordinate. Let χ ∈ (p k ) ⊥ red . Notice that if a ≡ a mod p k , then χ(a) = χ(a ); in other words, χ is well defined on O K /p k . Let ξ χ (g) = χ(g p k + p k ). It is clear that (U v ξ χ )(g) = χ(ι(v))ξ χ (g), i.e. ξ is an eigenfunction with eigenvalue χ(ι(v)). If χ(d k ) ⊥ red and d = p 1 · · · p s (distinct prime ideals), then χ = χ 1 · · · χ s , where χ i ∈ (p k i ) ⊥ red ; in this case the function ξ χ (g) = χ 1 (g p k 1 + p k 1 ) · · · χ s (g p k s + p k s ) is an eigenfunction for U v with eigenvalue χ(ι(v)). Since characters are orthonormal with respect to the Haar measure on G, we have that the discrete group {χ • ι} χ∈Λ ⊆ Z d = T d is the spectrum of the action Z d (G, Haar) and is clearly isomorphic to Λ.

More Formulae for the Correlation Functions
The goal of this section is to prove three results that we will use later. The first one (Proposition 6.1) is a generalization of a theorem by R.R. Hall [4]. Let η ∈ O K be the trivial character, η(a) = 1 for every a ∈ O K . For every r 1 and every a 1 , . . . , a r ∈ O K we have c r+1 (a 1 . . . , a r ) = a 0 a 1 · · · ar g(a 0 )g(a 1 ) · · · g(a r ) The following lemmata follow from Proposition 6.1, and deal with averages of the second and the third correlation functions, weighted by characters. These results are used in Section 7 when studying the spectral properties of the action O K (X, Π). Let us also point out that the proofs of Lemmata 6.2 and 6.3 are considerably simpler than the proofs of the analogous results in [3].
Before discussing the proofs of the two lemmata above, let us give the Proof of Proposition 6.1. Since χ 0 (0) = 1 it will be omitted in proof of (36). We use notation a = (a 0 , . . . , a r ) as in Section 4. Notice that the inner sum in (36) does not exceed in absolute value. Moreover, for every ideal a, |g(a)| 1 N (a k ) and the series a 0 a 1 · · · ar 1 N (lcm(a k ) converges absolutely. Let us evaluate the inner sum in (36). Let a = lcm(a k ). Notice that 1 N (a k ) a∈O K /a k χ 0 (a)χ 1 (a) · · · χ r (a) = 1 if χ 0 χ 1 · · · χ r = η; 0 otherwise.
This allows us to rewrite the inner sum in (36) as where a 0 = 0 and a j b j denotes the unique ideal c j such that a j = b j c j . Observe that and thus the inner sum in (36) equals Notice that the b i 's are necessarily square-free and thus µ(a i /b i ) = µ(a i )µ(b i ). Let us also observe that, for i = 0, . . . , r, To see this, for µ 2 (b i ) = 1, we can write the LHS of (40) as , since the products combined give the Euler product for ζ K (k). Alternatively, one can expand the products into sums and match terms with the series defining ζ K (k). Now (39) and (40) imply that the multiple sum in (36) equals and by Lemma 4.9 we get the desired statement.
Proof of Lemma 6.2. Observe that By Proposition 6.1 Proof of Lemma 6.3. Using (41) and Proposition 6.1 the LHS of (38) can be written as a 0 ,a 1 ,a 2 g(a 0 )g(a 1 )g(a 2 ) Consider the space X = {0, 1} O K , whose elements are O K -indexed sequences x = (x(a)) a∈O K , equipped with the Borel σ-algebra generated by cylinder sets. Introduce on X the probability measure Π defined as follows: for every r 0 and every a 0 , a 1 , . . . , a r ∈ O K , where c r+1 is the (r + 1)-st correlation function (10) associated to F k . It is clear that (42) determines the measure Π uniquely. We call Π the natural measure corresponding to the set of k-free integers in O K . If we consider the O K -action on X defined as b · x = (x(a + b)) a∈O K for b ∈ O K and x ∈ X, then it follows immediately from (42) that Π is invariant under this action. We can now reformulate the main result of this paper, of which Theorem 1.1 is a simplified version. For a ∈ O K , let U a be the unitary operator on H = L 2 (X, Π) given by The proof of Theorem 7.1-(i) requires us to show that there exists an orthonormal basis {θ χ } χ∈Λ for L 2 (X, Π) such that U a θ χ = χ(a)θ χ . First we will show that Λ is contained in the spectrum of the action O K (X, Π). For χ ∈ Λ, let us define the function θ χ : X → C, Proposition 7.2. Let χ ∈ Λ. Then (43) defines a function θ χ ∈ H, satisfying for Π-almost every x ∈ X.
Proof. Let f 0 ∈ H, f 0 (x) = x(0), and for a ∈ O K let U a,χ be the unitary operator on H defined by (U a,χ f )(x) = χ(−a)f (a · x).
For a ∈ O K let us denote by x(a) the function X → {0, 1} given by the projection of x ∈ X onto its a-th coordinate. We have the Proof. It is enough to show that the inner products x(a), θ χ , for a ∈ O K , are in general nonzero. We actually prove something more, that is an explicit formula for these inner products. Let χ ∈ (d k ) ⊥ red . We claim that for every a ∈ O K we have where g is the function defined in Proposition 6.1. To see this, observe that x(a), . From (43) and Lemma 6.2 we get Propositions 7.2 and 7.3 show that Λ is contained in the spectrum of the action O K (X, Π). Notice that, since U a is a unitary operator for every a ∈ O K , the eigenfunctions θ χ are orthogonal to one another for different χ ∈ Λ. Introduce the distinguished subspace H ⊆ H, Proof. By orthogonality, it is enough to show that the eigenfunctions span the space of all linear combinations of the x(a)'s. Let us show that H is isomorphic to L 2 ( O K , ν), where ν is the spectral measure.
Let us denote the the normalized eigenfunctions byθ χ = θ χ / θ χ . If we write x(a) = χ∈Λ x(a),θ χ θ χ , then we can retrieve the fact that x(a) = 1 using Proposition 7.4 and Lemma 7.5: The same argument allows us to provide an approximation of the function x(a) for a ∈ O K : let D 1 and define x D (a) = x(a),θ χ θ χ .
We have the following estimate for every ε > 0. Another important step in the proof of the Main Theorem is to show that the pointwise product of two eigenfunctions is still an eigenfunction. This is a peculiarity of actions with pure-point spectrum.
Proof. It is enough to show that for every a ∈ O K we have θ χ 1θ χ 2 , x(a) = ε θ χ , x(a) .
Therefore ε = θ χ 1θ χ 2 , x(a) θ χ , x(a) So far, we have proven that the family of eigenfunctions {θ χ } χ∈Λ is an orthonormal family for the subspace H, and that these eigenfunction have a remarkable multiplicative property. Now we want to show that the subspace H coincides with the full Hilbert space H. This will imply that {θ χ } χ∈Λ is in fact an orthonormal basis for H and therefore there is no "room" for other eigenspaces. In other words, Λ gives all the spectrum. This fact, together with Propositions 7.2 and 7.3, yields part (i) of Theorem 7.1. Theorem 1.1 (i) follows immediatley since uniqueness in (42) is guaranteed by Kolmogorov consistency. Finally, Proposition 5.4 and Mackey's theorem [11] imply that the two actions O K (X, Π) and Z d (G, Haar) are isomorphic. This constitutes part (ii) of Theorem 7.1, which gives Theorem 1.1 (ii).