Finitely additive measures on groups and rings

Sophie Frisch, Milan Paštéka, Robert F. Tichy and Reinhard Winkler

Abstract On arbitrary topological groups a natural finitely additive measure can be defined via compactifications. It is closely related to Hartman’s concept of uniform distribution on non-compact groups (cf. [Ha]). Applications to several situations are possible. Some results of M. Paštéka and other authors on uniform distribution with respect to translation invariant finitely additive probability measures on Dedekind domains are transfered to more general situations. Furthermore it is shown that the range of a polynomial of degree ≥ 2 on a ring of algebraic integers has measure 0.

1 Introduction

A sequence x = (x_{n})_{n∈N} of integers x_{n} ∈ Z is called uniformly distributed if it is uniformly distributed mod m on each of the finite rings Z_{m} = Z/(m),

(m) = mZ. To this concept there corresponds a finitely additive measure on a certain subsystem of the power set of Z. It is the completion of the system of finite unions of remainder classes of the form k +(m) with respect to the finitely additive probability measure μ generated by the requirement μ(k+(m)) = m^{−1}. There is a vast literature on generalizations of these concepts to more general classes of rings R (cf. references). Instead of (m) ⊆ Z one considers ideals I ⊴ R (I ⊲R if the inclusion is strict) with finite index #R/I and assigns to the classes r + I the measure (or norm) (#R/I)^{−1}, cf. [P1], [P2], [P-T1].

Here we continue such investigations but take the following point of view. To each family I_{j}, j ∈ J, of ideals with finite index (norm) there corresponds an embedding

ι : R → ∏_{j∈J} R/I_{j}, r ↦ (r + I_{j})_{j∈J},

of R into a compact topological ring. Let C = \overline{ι(R)} be the topological closure of ι(R) which is again a compact topological ring. C has a natural measure theoretic structure given by the Haar measure μ on the compact group (C, +). For the theory of uniform distribution the suitable concept of measurable sets is that of μ-continuity sets M which are defined by the property μ(∂M) = 0 for their topological boundary ∂M. The preimages ι^{−1}(M) of μ-continuity sets M form a set algebra (in general not a σ-algebra) which coincides with the system of measurable sets from the first paragraph if all ideals with finite index occur in the product. This is one part of Theorem 4 which also asserts that the measures induced by both constructions coincide.

Note that the embedding ι is not necessarily injective. As an example take any infinite field F, where the only ideal with finite index is F itself. Hence the embedding is the trivial map onto the one element ring and the concept is trivial.

By Pontrjagin’s duality (for the structure theory of locally compact abelian groups cf. [He-R]) this cannot happen if one considers compactifications of the additive group. In terms of projective properties one may compare compactifications in such a way that there is a maximal compactification, the well known Bohr compactification. Among all compactifications the Bohr compactification gives the maximal system of measurable sets. Theorem 1 shows that compactifications, measures and measurability fit together in a natural way. All these ideas are carried out in section 2.

In section 3 the concept is compared with other approaches.

In Theorem 2 we show that in the case that R is the ring of integers each set which is measurable in the sense of compactifications has a density equal to its measure. The converse is not true as Theorem 3 implies. There are sets having a density which are not measurable.

Theorems 4, 5 and 6 are devoted to the equivalence of the approaches via ideal measures and via compactifications for rings with unity. We conclude section 3 with some examples, especially with a detailed investigation of completion of a ring with unity with respect to a natural metric. This is closely connected with questions concerning uniformly distributed sequences in such rings. In the final section 4 the ideal measures of special sets are computed. This includes the result that the range of a polynomial of degree at least 2 on an algebraic ring of integers has measure 0.

Our results are related to invariant means on topological groups. A deeper understanding of the connection of both approaches should be the aim of investigations in the future.

As a general agreement we suppose all topological spaces to satisfy the Hausdorff separation axiom.

2 Compactifications and measures

2.1 Several notions and facts on compactifications

In the following we present a brief outline on compactifications of topological groups. (Generalizations to more general algebraic structures are possible.) For a more detailed description of the constructions see for instance [D-Pr-S], page 71. Note the analogies with arbitrary compactifications of completely regular topological spaces, especially with the Stone-Čech compactification which is the maximal one.

Let G be any topological group. A pair (C, ι) is called compactification of G if ι : G → C is a continuous homomorphism, C is a compact group and ι(G) is dense in C. (In general topology one often requires that ι is a homeomorphic imbedding.) The compactification is called injective if ι is injective. If one likes, one can force injectivity by considering G/ ker ι instead of G. (C_{1}, ι_{1}) is called smaller than (C_{2}, ι_{2}) (we write (C_{1}, ι_{1}) ≤ (C_{2}, ι_{2}) via φ) if there is a continuous epimorphism φ : C_{2} → C_{1} such that φι_{2} = ι_{1}. The relation ≤ is reflexive and transitive. Hence there is a natural notion of equivalence of compactifications: (C_{1}, ι_{1}) and (C_{2}, ι_{2}) are called equivalent if there exist φ and ψ such that (C_{1}, ι_{1}) ≤ (C_{2}, ι_{2}) via φ and (C_{2}, ι_{2}) ≤ (C_{1}, ι_{1}) via ψ. In this case ψ = φ^{−1}. (Obviously this could also be expressed in terms of categories.)

By standard cardinality arguments on appropriate systems of filters on G the equivalence classes of compactifications of G can be represented by a set. Hence we are allowed to fix a set C which contains exactly one representative of each equivalence class of compactifications of G. Any compactification may be identified with its equivalent copy in C.

The relation ≤ is a partial order on C. But far more is true: If (C_{i}, ι_{i}), i ∈ I, is any family of compactifications we may consider the direct product. Let P = ∏_{i∈I} C_{i} and let the continuous homomorphism ι : G → P defined by g ↦ (ι_{i}(g))_{i∈I}. If C is the closure of ι(G) in the compact group P then (C, ι) turns out to be the least common upper bound of all (C_{i}, ι_{i}). Note that the trivial compactification onto the one element group is the least upper bound of the empty set. Thus (C, ≤) in fact is a complete lattice. (Since we do not need this fact in full generality we do without the somewhat tedious proof.) In particular there is a maximal compactification, called the Bohr compactification (bG, bι). It follows from Pontrjagin’s duality that, if G is a discrete abelian or, more general, a locally compact abelian group, the Bohr compactification is

injective and can be obtained by taking the dual ˆ Ĝ_{d} of the discretely topologized dual Ĝ_{d} of G.

For us there is a second compactification of special interest. If one considers compactifications (C, ι) of the additive group (R, +) of a ring it might be convenient to extend the ring multiplication from R to C in a continuous way. If this is possible the continuation of ring multiplication (as addition) is, by density, uniquely determined by ι. We then call (C, ι) a ring compactification. The maximal ring compactification - we denote it by rbR - may happen to be not injective as the extreme example of an infinite field F shows where rbF = {0}. This follows from the fact that finite fields are the only compact topological rings that are fields, because compact rings with identity have an ideal topology, cf. [Wr] 32.3 and 32.5. This means that there is a neighbourhood base for 0 ∈ R consisting of clopen (= closed and open) ideals.

2.2 Compactifications and finitely additive measures

On every compact group C there is a unique regular probability measure μ_{C} which is a complete Borel measure and which is left and right invariant. μ_{C} is called the Haar measure. In our context this gives rise to the following notions:

Let G be any topological group and (C, ι) a compactification of G. Then we call a subset T ⊆ G measurable with respect to (C, ι) if it is the preimage under ι of a μ_{C}-continuity set M, i.e. T = ι^{−1}(M) with μ_{C}(\overline{M} ∩\overline{(C \ M)}) = 0. Thus a set M ⊆ C is a μ_{C}-continuity set if and only if its topological boundary ∂M is a zero set. It is easy to check that the systems S_{C} of all μ_{C}-continuity sets in C and the system S_{(C,ι)} of their preimages T ⊆ G – which we call the sets measurable with respect to (for short w.r.t.) (C, ι) – form a set algebra (not necessarily a σ-algebra) on C resp. on G. Obviously every T ∈ S_{(C,ι)} satisfies T = ι^{−1}(ι(T)). Note that T ∈ S_{(C,ι)} implies that \overline{ι(T)} ∈ S_{C} but that the converse is not true. (Consider for instance the compact group G = C = R/Z, ι the identity and T a dense set which is not a continuity set.) Considering the open kernels of \overline{ι(T)} and of its complement which, together, form the complement of its boundary, we observe for T = ι^{−1}(ι(T)): T ∈ S_{(C,ι)} if and only if there are disjoint open sets O_{1}, O_{2} ⊆ C with ι(T)∩O_{2} = ∅, ι(G\T)∩O_{1} = ∅ and μ_{C}(O_{1})+μ_{C}(O_{2}) = 1. The definition μ_{(C,ι)}(T) = μ_{C}(\overline{ι(T)}) or, equivalently, μ_{(C,ι)}(ι^{−1}(M)) = μ_{C}(M), transfers the measure μ_{C} on S_{C} to the system S_{(C,ι)}, thus defines the natural finitely additive measure on G w.r.t. (C, ι), defined for all T ∈ S_{(C,ι)}. A similar situation is investigated in the papers of Paštéka.

If the compactification is not injective then the measure μ_{(C,ι)} is in general not complete. Nevertheless, if μ_{C} is complete, the following similar statement holds: If the family T_{i} ∈ S_{(C,ι)}, i ∈ I, satisfies inf_{i∈I} μ_{(C,ι)}(T_{i}) = 0, then every T with T = ι^{−1}(ι(T)) which is contained in the intersection of the T_{i} is in S_{(C,ι)} and has measure 0. To see this, take closed sets M_{i} ∈ S_{C} whose preimages are the T_{i}. The intersection of the M_{i} is a closed set M of measure 0 which contains ι(T). Thus ι(T) ∈ S_{C}, since μ_{C} is complete, implying T = ι^{−1}(ι(M)) ∈ S_{(C,ι)} with measure 0.

The maximal compactification (Bohr compactification) (C, ι) = (bG, bι) of G will play a special role. Hence we write μ_{G} for μ_{(C,ι)}.

2.3 Compatibility of compactifications and measures

As expected, the measure μ_{(C,ι)}(T) of a set T ⊆ G does not depend on the compactification (C, ι) in the following sense:

Theorem 1 Let (C_{1}, ι_{1}) and (C_{2}, ι_{2}) be compactifications of G. On the intersection S_{(C_{1},ι_{1})} ∩ S_{(C_{2},ι_{2})} the measures μ_{1} = μ_{(C_{1},ι_{1})} and μ_{2} = μ_{(C_{2},ι_{2})} coincide. (C_{1}, ι_{1}) ≤ (C_{2}, ι_{2}) via φ implies S_{(C_{1},ι_{1})} ⊆ S_{(C_{2},ι_{2})}. In this case, M ∈ S_{C_{1}} implies φ^{−1}(M) ∈ S_{C_{2}}.

Proof: We proceed in several steps. Under the additional assumption (C_{1}, ι_{1}) ≤ (C_{2}, ι_{2}) via φ we prove the following facts.

1. \overline{φ^{−1}(M)} ⊆ φ^{−1}(\overline{M}) for every M ⊆ C_{1}: Continuity of φ.

2. ∂(φ^{−1}(M)) ⊆ φ^{−1}(∂(M)) for all M ⊆ C_{1}: Using the first fact we observe

\begin{aligned}∂(φ^{−1}(M)) & = \overline{φ^{−1}(M)} ∩ \overline{C_{2} \ φ^{−1}(M)} ⊆ φ^{−1}(\overline{M}) ∩ φ^{−1}(\overline{C_{1} \ M}) = \\ & = φ^{−1}(\overline{M} ∩ \overline{C_{1} \ M}) = φ^{−1}(∂M). \end{aligned}

3. μ_{C_{2}}(φ^{−1}(M)) = μ_{C_{1}}(M) for every measurable set M ⊆ C_{1}: The left hand side of the equality defines a translation invariant probability measure on the Borel sets of C_{1}, hence has to coincide with the unique Haar measure μ_{C_{1}}.

4. M ∈ S_{C_{1}} implies φ^{−1}(M) ∈ S_{C_{2}}: The assumption means μ_{1}(∂(M)) = 0, hence by 2. and 3.

μ_{C_{2}}(∂(φ^{−1}(M)) ≤ μ_{C_{2}}(φ^{−1}(∂(M))) = μ_{C_{1}}(∂(M)) = 0.

5. S_{C_{1},ι_{1}} ⊆ S_{C_{2},ι_{2}}: By the criterion in Section 2.2 T ∈ S_{(C_{1},ι_{1})} implies that there are disjoint open sets O_{1}, O_{2} ⊆ C_{1} with ι_{1}(T) ∩ O_{2} = ∅ = ι_{1}(G \ T) ∩ O_{1} and μ_{C_{1}}(O_{1}) + μ_{C_{1}}(O_{2}) = 1. Using 4. we get that O′_{i} := φ^{−1}(O_{i}), i = 1, 2, play the same role in C_{2}. ker ι_{2} ⊆ ker φι_{2} = ker ι_{1} and ι_{1}^{−1}ι_{1}(T) = T implies ι_{2}^{−1}ι_{2}(T) = T Thus, again by the same criterion, we conclude T ∈ S_{(C_{2},ι_{2})}.

6. T ∈ S_{(C_{1},ι_{1})} implies μ_{1}(T) = μ_{2}(T): By 5. T ∈ S_{(C_{2},ι_{2})}. Hence for both values i = 1, 2 we have the relation

μ_{C_{i}}(\overline{ι_{i}(T)}) + μ_{C_{i}}(\overline{ι_{i}(G \ T)}) = 1.

Using 1. we get

\overline{ι_{2}(T)} ⊆ \overline{φ^{−1}φι_{2}(T)} ⊆ φ^{−1}(\overline{φι_{2}(T)}) = φ^{−1}(\overline{ι_{1}(T)}),

hence by 3. μ_{C_{2}}(\overline{ι_{2}(T)}) ≤ μ_{C_{1}}(\overline{ι_{1}(T)}). The same holds for G \ T instead of T which, together with the above relations, is possible only if μ_{1}(T) = μ_{2}(T).

We have proved everything for the case (C_{1}, ι_{1}) ≤ (C_{2}, ι_{2}). The general case follows since two compactifications have a common upper bound, for instance (bG, bι_{G}) with measure μ_{G}:

μ_{1}(T) = μ_{G}(T) = μ_{2}(T) q.e.d.

Theorem 1 has the following consequence for further investigations. If a set T ⊆ G is measurable w.r.t. any compactification (C, ι) then it is measurable w.r.t. all bigger compactifications. The value μ_{(C,ι)}(M) does not dependent on the compactification as long as T is measurable w.r.t. it. Thus the Bohr compactification and the corresponding finitely additive probability measure μ_{G} = μ_{(bG,bι)} tells us everything about measures of sets T ⊆ G. Let us call μ_{G} the Hartman measure on G, cf. [Ha], and the corresponding measurable sets T ⊆ G group measurable or Hartman measurable. If we replace the Bohr compactification by the ring Bohr compactification we call the measurable sets ring measurable.

We are interested in measurability properties of subsets T ⊆ G. The concepts are nontrivial since it is possible to construct sets T ⊆ G which are not Hartman measurable, which follows from Theorem 3. Furthermore there are Hartman measurable sets that are not ring measurable. Consider for instance a ring which is an infinite discrete field. In this case Bohr and ring Bohr compactification are not equivalent in an extreme way.

3 Comparing several concepts

3.1 Sets of integers: Hartman measurability and density

In the special case R = Z one has a further nontrivial and natural finitely additive measure, the density. For a set T ⊆ Z of integers consider, for any finite set S ⊆ Z, the number A(T, S) = \frac{#T ∩S}{#S}. If, for the sets S_{N} = \{n ∈ N | n ≤ N\} and −S_{N} = {−n | n ∈ S_{N}}, both sequences A(T, S_{N}) and A(T, −S_{N}) tend, for N → ∞, to the same limit, we denote this common limit by dens(T) and call it the density of T.

It turns out that every Hartman measurable set has a density coinciding with its Hartman measure. The converse is not true, since the density of Hartman measurable sets is even uniform in the following sense:

Let T ⊆ Z be a set of integers. We say that T has the (unique) uniform density dens(T) if the following holds: For every ε > 0 there is a positive integer N_{ε} such that every set I_{k_{1},k_{2}} = {n ∈ Z | k_{1} < n ≤ k_{2}} with k_{2} − k_{1} ≥ N_{ε} fulfills

dens(T) − ε ≤ A(T, I_{k_{1},k_{2}}) ≤ dens(T) + ε.

Of course if the uniform density exists then also the density of T exists and both values are equal.

Theorem 2 Every Hartman measurable set T ⊆ Z of integers has a uniform density dens(T) with dens(T) = μ_{G}(T).

Proof: We consider the Bohr compactification (C, ι) = (bZ, bι) which can be realized by

ι(k) = (kα)_{α∈R/Z} ∈ C ⊆ ∏_{α∈R/Z}(\overline{χ_{α}(Z)}, χ_{α}).

Here every character χ_{α} : G → R/Z, k ↦ kα, of the integers represents a compactification. A topological base B is given by all sets B ⊆ C of the type

B = B(α_{1}, …, α_{k}, I_{1}, …, I_{k}) = {(x_{α})_{α∈R/Z} ∈ C | x_{α_{j}} ∈ I_{j}, j = 1, …, k}, where k ∈ N, α_{j} ∈ R/Z, and the I_{j} are connected open subsets (intervals) in R/Z. Note that all these base sets are μ-continuity sets where μ is the Haar measure on C. The same is true if we consider the base sets of a smaller compactification (instead of (C, ι)) where not all α ∈ R/Z occur.

If T ∈ S = S_{(C,ι)} then μ(∂M) = 0 where M = \overline{ι(T)}. Given any ε > 0 the outer regularity of μ gives an open set V ⊇ ∂M with μ(V) < ε/2. For each x ∈ ∂(M) take an open neighbourhood B_{x} ⊆ V which is a base set B_{x} ∈ B. Use compactness to get a finite subcovering of ∂M by open base sets B_{i}, i.e.

∂M ⊆ ⋃_{i=1}^{n_{1}} = U ⊆ V.

U is an open μ-continuity set. M_{0} = M \ U is compact and, similarly, can be covered by a finite union of base sets in such a way that

M_{0} = ⋃_{i=n_{1}+1}^{n_{2}} B_{i} ⊆ M ∪ U.

Thus we have M_{0} ⊆ M ⊆ M_{1} = M_{0} ∪ U with μ-continuity sets M_{i} which are finite unions of base sets B_{i}.

The definition of the involved sets uses only finitely many B_{i}, each of them involving only finitely many α_{j}, j = 1, …, s. Thus we may consider the projection π : C → C_{ε} = \overline{π(C)}, ι_{ε} = πι, onto the occuring components corresponding to α_{j}, j = 1, …, s. Hence (C_{ε}, ι_{ε}) is a finite dimensional compactification of Z, generated by the characters χ_{α_{j}}, j = 1, …, s. This means that C_{ε} ⊆ (R/Z)^{s} is (in the topological sense) generated by the element α = (α_{1}, …, α_{s}) ∈ C_{ε}. Write A′ = π(A) for arbitrary A ⊆ C. Since all B_{i} depend only on the components corresponding to the α_{j}, j = 1, …, s, we have π^{−1}(B′_{i}) = B_{i} for i = 1, …, n_{1} + n_{2}. This implies π^{−1}(U′) = U and π^{−1}(M′_{i}) = M_{i} for i = 0, 1. With T_{i} = ι^{−1}(M_{i}) we conclude

μ_{G}(T_{0}) = μ_{C_{ε}}(M′_{0}) ≤ μ_{G}(T) ≤ μ_{G}(T_{1}) = μ_{C_{ε}}(M′_{1}).

It is known from the theory of uniform distribution on monothetic groups (cf. [K-N], p. 269, Corollary 4.1) that the sequence (kα) is well distributed in C_{ε} for every generating element α. In our case this means that there is a positive integer N_{ε} such that for all k ∈ Z and all N ≥ N_{ε} we have

μ_{C_{ε}}(M′_{i}) − ε/2 < \frac{1}{N} #\{n ∈ Z | k < n ≤ k + N, nα ∈ M′_{i}\} < μ_{C_{ε}}(M′_{i}) + ε/2

for i = 0, 1. Furthermore we have

μ_{C_{ε}}(M′_{1}) ≤ μ_{C_{ε}}(M′_{0}) + μ_{C_{ε}}(U′) < μ_{C_{ε}}(M′_{0}) + \frac{ε}{2}.

Thus we get for every I = I_{k_{1},k_{2}} with k_{2} − k_{1} ≥ N_{ε}

\frac{#I ∩ T}{#I} ≤ \frac{#I ∩ T_{1}}{#I} < μ_{C_{ε}}(T_{1}) + \frac{ε}{2} < μ_{C_{ε}}(M′_{0}) + ε < μ_{G}(T) + ε,

and similarly \frac{#I∩T}{#I} ≥ μ_{G}(T) − ε, proving the theorem. q.e.d.

As a corollary we get that sets with density are not necessarily measurable.

Theorem 3 There are sets of integers which have a density but are not Hartman measurable.

Proof: Since the density of a Hartman measurable set is uniform by Theorem 2, it suffices to construct a set T which has a density but not a uniform density. Take T_{k} = {k^{2}, k^{2} + 1, …, k^{2} + k}, T^{+} = ⋃_{k∈N} T_{k}, T^{−} = {−n | n ∈ T^{+}} and T = T^{+} ∪ T^{−} then it is easy to check that T has the density dens(T) = 1/2 which is not uniform. q.e.d.

3.2 Compactifications and families of ideals with finite index

In this subsection we show that the approach via ring compactifications is equivalent to that via ideals of finite index.

For an arbitrary topological ring R with identity let J = {I_{j} | j ∈ J} be a family of clopen ideals I_{j} ⊴ R of finite index #R/I_{j} and suppose that J is closed under intersections. It is known (cf. for instance [P1], [P5]) that J, if it is closed under finite intersections, defines a finitely additive measure μ_{J} on R in the following way:

For every subset T ⊆ R which is a finite union T = ⋃_{i=1}^{n} r_{i} + I_{j_{i}} with pairwise disjoint r_{i} + I_{j_{i}} the number μ_{J} (T) = ∑_{i=1}^{n} #R/I_{j_{i}} is independent of the representation of T. Let us call such sets J -definable. The set function μ_{J} is a finitely additive measure μ_{J} on the set algebra of J -definable sets. μ_{J} can be uniquely extended to the so-called ideal measure (induced by J) on the set algebra of all subsets T ⊆ R which can be approximated in the following sense:

T is called measurable w.r.t. J if there is a (unique) number μ_{J} (T) such that for each ε > 0 there are J -definable sets A_{ε}, B_{ε} ⊆ R with A_{ε} ⊆ T ⊆ B_{ε} and

μ_{J} (T) − ε < μ_{J} (A_{ε}) ≤ μ_{J} (B_{ε}) < μ_{J} (T) + ε.

We show that this approach essentially leads to the same concepts as the compactification (C_{J} , ι_{J}) defined by

ι_{J} : R → ∏_{j∈J} R/I_{j}, r ↦ (r + I_{j})_{j∈J}.

For notational convenience call J point separating if ⋂_{j∈J} I_{j} = {0}. Note that this is the case if and only if ι_{J} is injective if and only if T = ι_{J}^{−1}ι_{J} (T) for all T ⊆ R.

Theorem 4 Suppose T ⊆ R. If T ∈ S_{(C_{J} ,ι_{J})} then it is measurable w.r.t. J. The converse holds if T = ι_{J}^{−1}ι_{J} (T) = T. If both values μ_{J} (T) and μ_{(C_{J} ,ι_{J})}(T) are defined they coincide. Hence μ_{J} = μ_{(C_{J} ,ι_{J})} if J is point separating.

Proof: First note that the J -definable sets T ⊆ R correspond to clopen sets M_{T} = \overline{ι_{J} (T)} which form a topological base of C_{J}. A set is clopen if and only if it has empty topological border. Hence all these sets are measurable w.r.t. J as well as w.r.t. the compactification (C_{J} , ι_{J}). By translation invariance of the Haar measure it is obvious that μ_{J} (T) = μ_{(C_{J} ,ι_{J})}(T) for such sets.

Assume now that T is measurable w.r.t. the compactification (C_{J} , ι_{J}), then μ_{C_{J}} (∂M_{T}) = 0. By regularity of the Haar measure and a compactness argument as in the proof of Theorem 2 ∂M_{T} can be covered by a finite union U of clopen base sets which has arbitrary small measure. Again as in the proof of Theorem 2 U is defined by only finitely many components, i.e. ideals, hence it induces clopen base sets M_{0} ⊆ M_{T} ⊆ M_{1} approximating M. By construction the preimages T_{i} = ι_{J}^{−1}(M_{i}) are J -definable sets approximating T in the sense of the definition. This, together with the remark above, shows that the values of both measures coincide.

It remains to prove that every T with T = ι_{J}^{−1}ι_{J} (T) = T which is measurable w.r.t. J is in S_{(C_{J} ,ι_{J})}. Put M = \overline{ι_{J} (T)}. For each ε > 0, there are J -definable sets A_{ε} and B_{ε} with A_{ε} ⊆ T ⊆ B_{ε} and μ_{J} (B_{ε} \ A_{ε}) < ε. Note that ∂M ⊆ M_{B_{ε}} \ M_{A_{ε}}. Furthermore their Haar measure equals to the ideal measure. It follows immediately that μ_{C_{1}}(∂M) = 0. Thus T = ι_{J}^{−1}ι_{J} (T) = T is measurable with respect to (C_{J} , ι_{J}).

On the other hand, if T is measurable w.r.t. the compactification (C_{J} , ι_{J}), then μ_{C_{J}} (∂M_{T}) = 0. By regularity of the Haar measure and a compactness argument as in the proof of Theorem 2 ∂M_{T} can be covered by a finite union U of clopen base sets which has arbitrary small measure. Again as in the proof of Theorem 2 U is defined by only finitely many components, i.e. ideals, hence it induces clopen base sets M_{0} ⊆ M_{T} ⊆ M_{1} approximating M. By construction the preimages T_{i} = ι_{J}^{−1}(M_{i}) are J -definable sets approximating T in the sense of the definition. q.e.d. Remark: For each I ⊴ R d_{I}(r_{1}, r_{2}) = 0 if r_{1} − r_{2} ∈ I and d_{I}(r_{1}, r_{2}) = 1 otherwise defines a pseudometric d_{I} on R. For given J there corresponds the system of all d_{I} with I ∈ J. This system is point separating if and only if J is point separating. The completion with respect to the uniformity of this system of pseudometrics turns out to be equivalent to (C_{J} , ι_{J}). The construction and the proof is standard. If J is countable the system of pseudometrics can be replaced by a metric. This special case is discussed in detail in section 3.3.

It is clear that the class of J -measurable sets increases with J. This is an immediate consequence of the following theorem together with Theorem 1.

Theorem 5 J_{1} ⊆ J_{2} implies (C_{J_{1}}, ι_{J_{1}}) ≤ (C_{J_{2}}, ι_{J_{2}})

Proof: As one checks easily, the mapping

φ : ι_{J_{2}}(R) → ι_{J_{1}}(R), (r + I)_{I∈J_{2}} ↦ (r + I)_{I∈J_{1}}

is well defined and can be uniquely extended to a continuous epimorphism C_{J_{2}} → C_{J_{1}}. q.e.d.

Hence it remains to investigate the largest possible choice for J. Let F be the system of all ideals I ⊴ R with finite index. Note that F is closed under finite intersections (cf. [P1], [P2]). The situation is explained by the following theorem.

Theorem 6 (C_{F}, ι_{F}) as a compactification of R (R ring with identity) is equivalent to the ring Bohr compactification (C, ι) = (rbR, rbι).

Proof: Since (C_{F}, ι_{F}) is a ring compactification there is a continuous epimorphism φ : C → C_{F} from the maximal ring compactification C onto C_{F} with φι = ι_{F}. Since C is compact, every continuous injection into a Hausdorff space is a homeomorphism. Thus it suffices to prove ker φ = {0}.

Let U be any open neighbourhood of 0 ∈ C. It follows from the structure theory of compact rings with unity (cf. [Wr], Theorem 32.3 and 32.5) that C has a topological base of clopen ideals which, since C is compact, must have finite index. Let I ⊴ C be such a clopen ideal with I ⊆ U. It follows that I_{0} = ι^{−1}(I) ⊴ R with |R/I_{0}| = |C/I|. Hence I_{0} ∈ F and ker ι_{F} ⊆ I_{0}. We conclude

ι^{−1}(ker φ) = ker(ιφ) = ker ι_{F} ⊆ I_{0}

and thus

ker φ = ιι^{−1}(ker φ) ⊆ ι(I_{0}) = I ⊆ U.

Since this holds for an arbitrary neighbourhood U of 0 ∈ C and since we have required the Hausdorff separation axiom we have ker φ = {0}, proving the theorem. q.e.d.

Theorem 6 implies that μ_{(C_{F} ,ι_{F})} and μ_{(rbR,rbι)} are defined on the the same set algebra and therefore, by Theorem 1, coincide. We will call this finitely additive measure the ideal measure on the ring.

3.3 Examples

Let us investigate subsets T ⊆ Z of the integers and their measurability w.r.t. several compactifications. Take any α ∈ R and consider the compactification (C, χ_{α}) where C ≤ R/Z is the torus group and χ_{α} is the unique character with χ_{α}(1) = α + Z. If α = \frac{p}{q} is rational with integers q > 0 and p relatively prime, then the measurable sets are exactly the unions of classes w.r.t. the cyclic factor group Z/(q) modulo q. One gets the concept of uniform distribution modulo q.

If one looks at the supremum of all (C_{α}, ι_{α}) where α runs through all \frac{1}{p^{n}} where p is a prime number one gets the injective p-adic completion of the integers. All singletons are measurable zero sets. But also all p-powers form a measurable zero-set w.r.t. this compactification.

If n = \frac{1}{α} runs through all positive integers one gets the classical concept of uniform distribution in Z, cf. [Niv]. This compactification coincides with the ring Bohr compactification and is closely related to the Banach-Buck measure on the positive integers (cf. [B], [K-N] p. 313-315, [L-N], [Ok], [P1] and [P2]).

In the following we discuss in more detail the situation where R is a ring with 1 and J : I_{1} ⊇ I_{2} ⊇ I_{3} … ⊇ I_{n} ⊇ … is a sequence of ideals satisfying

⋂_{n=1}^{∞} I_{n} = {0}.

A metric d(x, y) = ∥x − y∥ can be introduced by the usual norm

∥x∥ = ∑_{n=1}^{∞} \frac{1 − χ_{I_{n}}(x)}{2^{n}},

where χ_{E} denotes the characteristic function of a set E. If J is uncountable one needs a system of pseudometrics inducing the corresponding uniform structure; cf. section 3.2.

Obviously d(x_{n}, y_{n}) → 0 if and only if for every N ∈ N there exists an n_{0} = n_{0}(N) such that for all n ≥ n_{0} the relation x_{n} ≡ y_{n} mod I_{N} holds. This yields that the ring operations are continuous. Denote by (Ω, d) the completion of the metric space (R, d). (A standard construction as it is carried out in the last section, paragraph 170, in the second volume of van der Waerden’s book [vdW] shows that ring operations can uniquely extended to Ω continuously.) Denote by \overline{S} the closure of a set S in Ω. Then the following elementary properties can be established.

(i) For any ideal I in R the closure \overline{I} is an ideal in Ω. (ii) x + \overline{I_{n}} = \overline{x + I_{n}} for x ∈ R, n = 1, 2, . . .. (iii) For every α ∈ Ω there exists an x ∈ R such that α + \overline{I_{n}} = x + \overline{I_{n}}. (iv) For all α ∈ Ω, n = 1, 2, … the set α + \overline{I_{n}} is open and closed.

(v) (x + \overline{I_{n}}) ∩ R = x + I_{n} for all x ∈ R, n = 1, 2, . . .. (vi) \overline{S} = ∩_{n=1}^{∞}(S + \overline{I_{n}}) for every S ⊆ Ω. (vii) The system \{x + \overline{I_{n}} | x ∈ Ω, n = 1, 2, …\} is a closed open base in Ω. (viii) Ω is compact if and only if each factor ring R/I_{n} is finite.

In the following we suppose that each factor ring is finite, i.e. Ω is compact. Following the general approach of section 2 we consider the group (Ω, +) with Haar measure μ_{Ω}. Furthermore we define for arbitrary S ⊆ R a set function μ̄_{J} by μ̄_{J} (S) = μ_{Ω}(\overline{S}). μ̄_{J} is an outer measure and is called covering density induced by J, and it can easily be seen that for A, B ⊆ R

μ̄_{J} (A ∪ B) + μ̄_{J} (A ∩ B) ≤ μ̄_{J} (A) + μ̄_{J} (B).

It follows from basic measure theory that the system

D_{J} = {S ⊆ R | μ̄_{J} (S) + μ̄_{J} (R \ S) = 1}

is a set algebra and the restriction of μ̄_{J} on D_{J} is a finitely additive measure on D_{J}. Denoting by [S : I_{n}] the number of different residue classes s + I_{n} with s ∈ S and putting N(I_{n}) = #R/I_{n} we clearly have

μ_{Ω}(α + \overline{I_{n}}) = \frac{1}{N(I_{n})}

for α ∈ Ω, n = 1, 2, . . .. Hence for S ⊆ R the covering density of S can be computed by the limit formula

μ̄_{J} (S) = lim_{n→∞} \frac{[S : I_{n}]}{N(I_{n})}.

Remark: Since every class α + \overline{I_{n}} is a μ_{Ω}-continuity set the basic notions of the abstract theory of uniform distribution of sequences can be applied to our situation, and general results in the flavour of [K-N], chapters 3,4,5 can be shown. For more recent results concerning distribution problems in rings and submeasures we refer to [P-T1]. For instance, a sequence (x_{n}) in R is called J -well distributed if and only if for each ideal I_{n} ∈ J and x ∈ R the relation

lim_{m→∞} \frac{1}{m}#\{k | h + 1 ≤ k ≤ h + m, x_{k} ≡ x mod I_{n}\} = \frac{1}{N(I_{n})}

holds uniformly in h = 1, 2, … (cf. also subsection 3.1). Following the ideas of [P-T1] we establish

Theorem 7 Let J be an ideal system as above and S ⊆ R with μ̄_{J} (S) = 1. Then a J -well distributed sequence can be selected from S.

Remark: Specific distribution results on linear recurring sequences in Dedekind domains are shown in [Ti-Tu1] and [Ti-Tu2].

4 Special sets

In the last section we restrict our investigations to commutative rings with identity. Assume J_{1}, J_{2}, … to be a sequence of coprime ideals and put I_{n} = J_{1} ∩ … ∩ J_{n}. A set S ⊆ R is called multiplicative if [S : I ∩ J] = [S : I] · [S : J] holds for arbitrary coprime ideals I, J. From the Chinese Remainder Theorem we have N(I_{n}) = N(J_{1}) · . . . · N(J_{n}), hence the limit formula in section 3.3 for the computation of the covering density yields

μ_{J} (S) = ∏_{n=1}^{∞} \frac{[S : J_{n}]}{N(J_{n})}

for any multiplicative set in R.

As an example of a multiplicative set let us consider a mapping f : R → R such that for each ideal I we have f(x) ≡ f(y) mod I provided that x ≡ y mod I. Due to the Chinese Remainder Theorem the mapping f(x) + I ∩ J ↦ (f(x) + I, f(x) + J) is a bijection between R/(I ∩ J) and R/I ⊕ R/J, and so the image set f(R) is multiplicative. Therefore the image set of each polynomial in R[x] is multiplicative. Let R^{k} denote the set of k-th powers of elements of R. Then R^{k} is multiplicative, too.

Let J be a maximal ideal with finite norm N(J). Then the multiplicative group of the field R/J is cyclic and let g + J be a generator. The elements x^{k} + J, x ∉ J form a cyclic subgroup generated by g^{k} + J. The order of this element is \frac{N(J)−1}{(k,N(J)−1)} and we have [R^{k} : J] = \frac{N(J)−1}{(k,N(J)−1)} + 1. Thus we have shown

Theorem 8 Let J_{n}, n = 1, 2, …, be maximal ideals in R (commutative ring with identity). Then

μ_{J} (R^{k}) = ∏_{n=1}^{∞} (\frac{N(J_{n}) − 1}{(k, N(J_{n}) − 1)} + 1) · \frac{1}{N(J_{n})}.

Corollary 1: If (k, N(J_{n}) − 1) = 1, n = 1, 2, …, then μ_{J} (R^{k}) = 1. Corollary 2: If (k, N(J_{n}) − 1) > 1 for infinitely many n, then μ_{J} (R^{k}) = 0.

Our final result is devoted to the ideal measure of the image set f(R) for nonlinear polynomials. Note that the ideal measure is defined via all ideals of finite index as in section 3.2.

Theorem 9 Let R be the ring of algebraic integers in a number field and let f ∈ R[x] be a non-linear polynomial. Then f(R) is of ideal measure 0.

Proof: Let n = deg f. By a theorem of Niederreiter and Lo [N-L], there are infinitely many maximal ideals P in R such that f is not a permutation polynomial mod P i.e., the function induced by f on the (finite) residue field R/P is not bijective. For each such P of index [R : P] = q, the value set f(R) is contained in the union of at most q − \frac{q−1}{n} residue classes of P, by a theorem of Wan [Wn].

For different maximal ideals P_{1}, …, P_{k} of index [R : P_{i}] = q_{i}, the ideal measure of the set of elements of R that are for each i in one of N_{i} given residue classes mod P_{i} is ∏_{i=1}^{k} \frac{N_{i}}{q_{i}}, by the Chinese Remainder Theorem. Thus, if f is not a permutation polynomial mod P_{i} for i = 1, …, k then the image of f is contained in a set of ideal measure at most

∏_{i=1}^{k}(1 − \frac{q_{i} − 1}{nq_{i}}) ≤ (1 − \frac{1}{2n})^{k}.

This value can be made arbitrarily small by considering an infinite sequence of different maximal ideals mod which f is not a permutation polynomial. q.e.d.

References

[B] R.C. Buck The measure theoretic approach to density, Amer. J. Math. 68 (1946), 560-580.

[D-Pr-S] D. N. Dikranjan, I. R. Prodanov and L. N. Stoyanov Topological Groups, Pure and Applied Mathematics, Marcel Dekker Inc., New York–Basel (1990).

[Ha] S. Hartman Remarks on equidistribution on non-compact groups, Comp. Math. 16 (1964), 66-71.

[He-R] E. Hewitt and K. A. Ross Abstract harmonic analysis, Springer (Second Edition 1979).

[K-N] L. Kuipers and H. Niederreiter Uniform distribution of sequences, J. Wiley and Sons, New York - London - Sydney - Toronto (1974).

[L-N] S.K. Lo and H. Niederreiter Banach-Buck measure, density, and uniform distribution in rings of algebraic integers, Pac. J. Math. 61, No. 1 (1975), 191-208.

[N-L] H. Niederreiter and S. K. Lo Permutation Polynomials over Rings of Algebraic Integers, Abh. Math. Sem. Hamburg 49 (1979) 126-139.

[Niv] I. Niven Uniform distribution of sequences of integers, Comp. Math. 16 (1964), 158-160.

[Ok] K.Okutsu On the measure on the set of positive integers, Proc. Japan Acad., 69 A, 173-174 (1993).

[P1] M. Paštéka The measure density on Dedekind domains, Ricerche di matematica.

[P2] M. Paštéka Submeasures and uniform distribution on Dedekind rings. Tatra Mountain Math. Publ.

[P-T1] M. Paštéka and R. F. Tichy Distribution problems in Dedekind domains and submeasures, Ann. Univ. Ferrara 40 (1994), 191-206.

[Ti-Tu1] R. F. Tichy and G. Turnwald Uniform distribution of recurrences in Dedekind domains, Acta Arith. 46 (1985), 81-89.

[Ti-Tu2] R. Tichy and G. Turnwald Weak uniform distribution of u_{n+1} = au_{n}+b in Dedekind domains, Manuscripta Math. 61 (1988), 11-22.

[Tu1] G. Turnwald Einige Bemerkungen zur diskreten Gleichverteilung, Wiener Seminarberichte 1984-1986, E. Hlawka (Ed.), Lecture Notes 1262, Zahlentheoretische Analysis II, 144-149.

[vdW] B. L. van der Waerden Algebra I+II, Springer, Berlin–Heidelberg–New York, (1967).

[Wr] S. Warner Topological rings, North Holland Mathematics Studies 178 (1993).

[Wn] D. Wan A p-adic Lifting Lemma and Its Applications to Permutation Polynomials, in “Finite Fields, Coding Theory, and Advances in Computing” (Proc. of Conf. in Las Vegas, Nevada 1991), G. L. Mullen and P. J.-S. Shiue eds., Dekker 1993.

S. Frisch and R.F. Tichy: Institut für Mathematik, TU Graz, Steyrergasse 30, A-8010 Graz, Austria.

M. Paštéka: Faculty of Sciences, University of Ostrava, Brafova 7, CZ-70103 Ostrava, Czech Republic.

R. Winkler: Institut für Algebra und Diskrete Mathematik, TU Wien, Wiedner Hauptstraße 8-10, A-1040 Wien; Kommission für Mathematik, Austrian Academy of Sciences, Dr. Ignaz Seipelplatz 2, A-1010 Wien; Austria.