Periodic decomposition of measurable integer valued functions

Tamás Keleti ^{∗}

April 4, 2007

Abstract

We study those functions that can be written as a sum of (almost everywhere) integer valued periodic measurable functions with given periods. We show that being (almost everywhere) integer valued measurable function and having a real valued periodic decomposition with the given periods is not enough. We characterize those periods for which this condition is enough.

We also get that the class of bounded measurable (almost everywhere) integer valued functions does not have the so called decomposition property. We characterize those periods a_{1}, …, a_{k} for which an almost everywhere integer valued bounded measurable function f has an almost everywhere integer valued bounded measurable (a_{1}, …, a_{k})-periodic decomposition if and only if Δ_{a_{1}} … Δ_{a_{k}}f = 0, where Δ_{a}f(x) = f(x + a) − f(x).

Introduction

In [8] those functions were studied that can be written as a sum of periodic integer valued functions with given periods a_{1}, …, a_{k}. Clearly these functions must be integer valued and they can be written as a sum of periodic real valued functions with given periods a_{1}, …, a_{k}. Several results were proved about the question whether the converse is true or false; that is, whether the existence of a real valued (a_{1}, …, a_{k})-periodic decomposition of an integer valued function implies the existence of an integer valued (a_{1}, …, a_{k})-periodic decomposition. Among others the following result were proved:

Theorem 0.1 ([8, Theorem 2.1]) Suppose that an integer valued function f : ℤ → ℤ can be written as f = g_{1} + … + g_{k}, where each g_{j} is a real valued a_{j}-periodic function for some a_{j} ∈ ℤ.

Then f can be also written as f = h_{1} + … + h_{k}, where each h_{j} is an integer valued a_{j}-periodic function.

∗Supported by Hungarian Scientific Foundation grants no. F 43620. and T 49786. This research started when the first author was a visitor at the Alfréd Rényi Institute of Mathematics of the Hungarian Academy of Science.

MSC code: 28A20; Secondary 39A10

Key Words: periodic functions, measurable functions, periodic decomposition, integer valued functions, almost everywhere integer valued functions, real valued functions, difference operator, decomposition property

The question whether the same result holds for functions defined on ℝ is still open:

Question 0.2 [8, Question 5.1] Is it true for any a_{1}, …, a_{k} ∈ ℝ that if an integer valued function f : ℝ → ℤ has a real valued (a_{1}, …, a_{k})-periodic decomposition then f also has an integer valued (a_{1}, …, a_{k})-periodic decomposition?

There are some positive partial results if we have some assumptions about the periods a_{1}, …, a_{k} (see [8] and [5]).

For bounded decomposition of bounded functions the following counterexample was found for the analogue question:

Theorem 0.3 ([8, Theorem 3.1]) There exists a function u : ℤ × ℤ → {0, 1} that can be written as a sum of a (0, 1)-periodic, a (1, 0)-periodic and a (1, 1)- periodic bounded ℤ × ℤ → ℝ function, can be written also as the sum of three periodic ℤ × ℤ → ℤ functions with the same periods but cannot be written as a sum of three periodic bounded ℤ × ℤ → ℤ functions with the same periods.

Note that by repeating this construction on each coset one can get a similar counter-example on any Abelian group that contains a ℤ × ℤ subgroup, so in particular there exist a_{1}, a_{2}, a_{3} ∈ ℝ and a function f : ℝ → {0, 1} that has a bounded real valued (a_{1}, a_{2}, a_{3})-periodic decomposition but does not have a bounded integer valued (a_{1}, a_{2}, a_{3})-periodic decomposition.

In the above cited results no measurability was assumed. In this paper we will study what happens if we allow only measurable functions. First we give a negative answer (Theorem 1.2) to the measurable analogue of Question 0.2 and then we characterize (Theorem 2.5) those periods for which we have positive result, at least if we are happy with almost everywhere integer valued decompositions. Everywhere integer valued measurable decompositions are studied in Section 3. It turns out that the question whether we can get integer valued decompositions instead of almost everywhere integer valued decompositions depends on the answers to the nonmeasurable questions like the above mentioned Question 0.2.

The characterization of those functions that can be written as a sum of periodic functions with given period has a much longer history. It started in the seventies with some unpublished work of I. Z. Ruzsa and continued among others in [1], [3], [5], [6], [7], [9], [10], [11], [12], [13], [14] and [15]. If f = f_{1} + … + f_{n} is an (a_{1}, …, a_{n})-periodic decomposition of f then

Δ_{a_{1}}Δ_{a_{2}} ⋯ Δ_{a_{n}}f = 0, where Δ_{a_{j}}f(x) = f(x + a_{j}) − f(x),  (1)

since the difference operators Δ_{a_{j}} commute. A class of functions F is said to have the decomposition property if every f ∈ F that satisfies (1) has an (a_{1}, …, a_{k})-periodic decomposition in F. Since for the identity function f(x) = x we clearly have Δ_{1}Δ_{1}f = 0 but it is not the sum of two 1-periodic functions, many natural classes of functions (e.g. all ℝ → ℝ functions, continuous ℝ → ℝ functions) do not have the decomposition property. However, many classes of functions do have the decomposition property: for example the class of all bounded continuous ℝ → ℝ functions [11], the class of all bounded ℝ → ℝ functions [3], [12], the class of all bounded measurable ℝ → ℝ functions [12] and the class of all bounded real valued functions on an arbitrary Abelian group [12].

For integer valued functions it was proved in [8] that the class of bounded ℤ → ℤ functions has the decomposition property but the class of bounded ℝ → ℤ functions does not have the decomposition property. In fact, among the torsion free Abelian groups only the additive subgroups of ℚ are those on which the class of bounded integer valued functions has the decomposition property [8, Corollary 3.5].

In this note we get (Corollary 1.3) that on ℝ the classes of bounded measurable (almost everywhere) integer valued functions and (almost everywhere) integer valued L_{∞} functions do not have the decomposition property. We characterize (Theorem 2.5) those periods a_{1}, …, a_{k} for which for any bounded measurable ℝ → ℤ function the existence of a bounded measurable almost everywhere integer valued (a_{1}, …, a_{k})-periodic decomposition is equivalent to (1). We show (Proposition 3.4) that this characterization is not valid for everywhere integer valued (a_{1}, …, a_{k})-periodic decompositions and finally we give a conjecture about a possible characterization of those periods a_{1}, …, a_{k} for which a bounded measurable integer valued function has a bounded measurable integer valued (a_{1}, …, a_{k})-periodic decomposition if and only if (1) holds.

Meanwhile, as a spin-off, we also characterize (Corollary 2.4) those periods a_{1}, …, a_{k} for which the measurable (a_{1}, …, a_{k})-periodic decomposition of an ℝ → ℝ/ℤ function is essentially unique. This characterization turns out to be different for ℝ → ℝ functions (Lemma 1.1).

1 A negative result

The following fact is known, see e.g. in [9].

Lemma 1.1 Let a_{1}, …, a_{k} ∈ ℝ \ {0} such that a_{i}/a_{j} ∉ ℚ for any i ≠ j and suppose that f_{1} +…+f_{k} = g_{1} +…+g_{k} and for each j, f_{j} and g_{j} are a_{j}-periodic measurable ℝ → ℝ functions.

Then f_{j} − g_{j} is almost everywhere constant for every j = 1, …, k.

It is easy to see that the condition a_{i}/a_{j} ∉ ℚ for i ≠ j is also necessary. We will see (Corollary 2.4) that for ℝ → ℝ/ℤ functions the necessary and sufficient condition for a_{1}, …, a_{k} is stronger.

The following theorem shows that the existence of a measurable real valued (a_{1}, …, a_{n})-periodic decomposition for an ℝ → ℤ function does not always implies the existence of a measurable integer valued (a_{1}, …, a_{n})-periodic decomposition, not even the existence of a measurable, almost everywhere integer valued (a_{1}, …, a_{n})-periodic decomposition.

Theorem 1.2 There exists an integer valued bounded Lebesgue measurable function on the real line that can be written as a sum of three real valued bounded measurable periodic functions but cannot be written as a sum of three almost everywhere integer valued measurable periodic functions with the same periods.

Proof. Let t ∈ ℝ \ ℚ be arbitrary and let

f(x) = {tx} + {(1 − t)x} + {−x},  (2)

where {.} denotes the fractional part; that is, {a} = a − [a]. Then f is clearly measurable and it is the sum of a \frac{1}{t}-periodic a \frac{1}{1−t}-periodic and a 1-periodic bounded measurable function. Noting that f can be also written as

f(x) = tx − [tx] + (1 − t)x − [(1 − t)x] − x − [−x] = −[tx] − [(1 − t)x] − [−x]

we get that f is integer valued.

Suppose that f = g_{1} + g_{2} + g_{3} and g_{1}, g_{2}, g_{3} are measurable almost everywhere integer valued periodic measurable functions with periods \frac{1}{t}, \frac{1}{1−t} and 1, resp. Since t ∉ ℚ and by adding a constant to {−x} we cannot get an almost everywhere integer valued function, applying Lemma 1.1 for {tx} + {(1 − t)x} + {−x} = g_{1} + g_{2} + g_{3}, we get a contradiction. □

Corollary 1.3 The following classes of functions do not have the decomposition property: {f : ℝ → ℤ | f ∈ L_{∞}}, {f : ℝ → ℤ | f is bounded and measurable}, {f : ℝ → ℝ | f ∈ L_{∞} and fis almost everywhere integer valued}, and {f : ℝ → ℝ | f is bounded, measurable and almost everywhere integer valued}.

Proof. Let t and f be as in the above proof. Then f is contained in all the above classes, Δ_{1}Δ_{1/t}Δ_{1/(1−t)}f = 0 but, as we saw it in the above proof, f cannot be written as a sum of a \frac{1}{1−t}-periodic, a \frac{1}{t} and a 1-periodic measurable almost everywhere integer valued function. □

2 Almost everywhere integer valued decompositions

The following lemma might be known but for completeness we present a proof.

Lemma 2.1 If E_{j} ⊂ ℝ is an a_{j}-periodic measurable set with positive measure for each j = 1, …, k and \frac{1}{a_{1}} , …, \frac{1}{a_{k}} are linearly independent over ℚ then E_{1} ∩ … ∩ E_{k} has positive measure.

Proof. By applying the Lebesgue’s Density Theorem for each E_{j}, we can find δ > 0 and d_{1}, …, d_{k} ∈ ℝ such that

λ((d_{j} − 2δ, d_{j} + 2δ) \ E_{j}) < \frac{2δ}{k} (j = 1, …, k).

For each j = 1, …, k, using that E_{j} is a_{j}-periodic, we get that for any m_{j} ∈ ℤ,

t ∈ (m_{j}a_{j} + d_{j} − δ, m_{j}a_{j} + d_{j} + δ) ⟹ λ ((t − δ, t + δ) \ E_{j}) < \frac{2δ}{k} . (3)

One form of Kronecker’s theorem (see e.g. in [4, Theorem 444]) states that if b_{1}, …, b_{k} ∈ ℝ are linearly independent over ℚ, c_{1}, …, c_{k} ∈ ℝ and ε > 0 then there exists t ∈ ℝ and m_{1}, …, m_{k} ∈ ℤ such that |b_{j}t − m_{j} − c_{j}| < ε for every j = 1, …, k.

Applying the above mentioned form of Kronecker’s theorem for b_{j} = \frac{1}{a_{j}} , c_{j} =

\frac{d_{j}}{a_{j}} and ε = \frac{δ}{a_{j}} we get a t ∈ ℝ such that |t − m_{j}a_{j} − d_{j}| < δ for every j. Then by (3),

λ ((t − δ, t + δ) \ E_{j}) < \frac{2δ}{k}

for every j = 1, …, k, which implies that λ(E_{1} ∩ … ∩ E_{k} ∩ (t − δ, t + δ)) > 0. □

We remark that Lemma 2.1 easily implies the following statement. In fact, the converse implication is also easy.

Corollary 2.2 If f_{1}, …, f_{k} : ℝ → (0, ∞) are periodic measurable functions such that the reciprocals of the periods are linearly independent over ℚ then

||f_{1} + … + f_{k}||_{∞} = ||f_{1}||_{∞} + … + ||f_{k}||_{∞}.

□

The following theorem shows that if \frac{1}{a_{1}} , …, \frac{1}{a_{k}} are linearly independent over ℚ then the almost everywhere integer valued measurable functions have only trivial measurable (a_{1}, …, a_{k})-periodic decompositions.

Theorem 2.3 Let a_{1}, …, a_{k} ∈ ℝ \ \{0\} such that \frac{1}{a_{1}} , …, \frac{1}{a_{k}} are linearly independent over ℚ. Suppose that f_{j} : ℝ → ℝ is an a_{j}-periodic measurable function for each j = 1, …, k and that f = f_{1} + … + f_{k} is an almost everywhere integer valued function.

Then each fractional part {f_{j}} is constant almost everywhere.

Proof. Let

F_{j} = ⋃ {f_{j}^{−1}((r, q)) : r, q ∈ ℚ, λ(f_{j}^{−1}((r, q))) = 0} (j = 1, …, k),

E = {x ∈ ℝ : f(x) ∈ ℤ} \ ⋃_{j=1}^{k} F_{j}.

Then λ(ℝ \ E) = 0, so it enough to prove that for any fixed u, v ∈ E and ε > 0 we have ||f_{1}(u) − f_{1}(v)|| < ε, where ||.|| denotes the distance from the nearest integer; that is, ||x|| = min({x}, {1 − x}).

Let

E_{1} = \{x ∈ ℝ : |f_{1}(x) − f_{1}(u)| < \frac{ε}{k} \}

and

E_{j} = \{x ∈ ℝ : |f_{j}(x) − f_{j}(v)| < \frac{ε}{k} \} (j = 2, 3, …, k).

For each j = 1, …, k the set E_{j} is measurable and a_{j}-periodic since f_{j} is measurable and a_{j}-periodic, and λ(E_{j}) > 0 since u, v ∈ E and so u, v ∉ F_{j}. Hence by Lemma 2.1, λ(E_{1} ∩ … ∩ E_{k}) > 0, so there exists a t ∈ E ∩ E_{1} ∩ … ∩ E_{k}.

Then

|f_{1}(u) − f_{1}(t)| < \frac{ε}{k} ,  (4)
|f_{j}(v) − f_{j}(t)| < \frac{ε}{k} (j = 2, …, k).  (5)

Since f_{1}(v) + f_{2}(v) + … + f_{k}(v) = f(v) ∈ ℤ and f_{1}(t) + f_{2}(t) + … + f_{k}(t) = f(t) ∈ ℤ, (5) implies that ||f_{1}(t) − f_{1}(v)|| < (k − 1)\frac{ε}{k}. Combining this with (4) we get that ||f_{1}(u) − f_{1}(v)|| < ε, which completes the proof. □

Now we can characterize those periods for which the measurable decomposition of an ℝ → ℝ/ℤ function is unique up to additive constants. Note that, by Lemma 1.1, the characterization is different for ℝ → ℝ functions.

Corollary 2.4 For any a_{1}, …, a_{k} ∈ ℝ \ {0} the following two statements are equivalent.

(i) If f_{1} + … + f_{k} = g_{1} + … + g_{k} and for each j, f_{j} and g_{j} are a_{j}-periodic measurable ℝ → ℝ/ℤ functions then f_{j} −g_{j} is almost everywhere constant for every j = 1, …, k.

(ii) \frac{1}{a_{1}} , …, \frac{1}{a_{k}} are linearly independent over ℚ.

Proof. (i) ⇒ (ii): Suppose that (ii) is false, so there exists (m_{1}, …, m_{k}) ∈ ℤ × … × ℤ \ {0, …, 0} such that

\frac{m_{1}}{a_{1}} + … + \frac{m_{k}}{a_{k}} = 0.

Then f_{j}(x) = \frac{m_{j}}{a_{j}} x mod 1 for j = 1, …, k and g_{1} = … = g_{k} = 0 shows that (i) is also false.

(ii) ⇒ (i): This follows simply from Theorem 2.3 □

Now we can characterize those periods for which the existence of a (bounded) measurable real valued (a_{1}, …, a_{k})-periodic decomposition of an integer valued or almost everywhere integer valued function implies the existence of a (bounded) measurable almost everywhere integer valued (a_{1}, …, a_{k})-periodic decomposition. We will get the same characterization for those periods for which an integer valued or almost everywhere integer valued bounded measurable function has a bounded measurable almost everywhere integer valued decomposition if and only if Δ_{a_{1}} … Δ_{a_{k}}f = 0, where Δ_{a}f(x) = f(x+a)−f(x).

Theorem 2.5 For any a_{1}, …, a_{k} ∈ ℝ \ {0} the following seven statements are equivalent.

(i)/(i′) If an everywhere/almost everywhere integer valued measurable function f on ℝ can be decomposed as f = f_{1} + … + f_{k} such that each f_{j} is an a_{j}-periodic measurable ℝ → ℝ function then f can be also decomposed as f = g_{1} + … + g_{k} such that each g_{j} is an a_{j}-periodic almost everywhere integer valued measurable function.

(ii)/(ii′) If an everywhere/almost everywhere integer valued bounded measurable function f on ℝ can be decomposed as f = f_{1} + … + f_{k} such that each f_{j} is an a_{j}-periodic bounded measurable ℝ → ℝ function then f can be also decomposed as f = g_{1} + … + g_{k} such that each g_{j} is an a_{j}-periodic almost everywhere integer valued bounded measurable function.

(iii)/(iii′) An everywhere/almost everywhere integer valued bounded measurable function f on ℝ can be decomposed as f = g_{1} + … + g_{k} such that each g_{j} is an a_{j}-periodic almost everywhere integer valued bounded measurable function if and only if Δ_{a_{1}} … Δ_{a_{k}}f = 0.

(iv) If B_{1}, …, B_{n} are the equivalence classes of {a_{1}, …, a_{k}} with respect to the relation a ∼ b ⇔ a/b ∈ ℚ, and b_{j} denotes the smallest common multiple of the numbers in B_{j} (for each j = 1, …, n) then \frac{1}{b_{1}} , …, \frac{1}{b_{n}} are linearly independent over ℚ.

Proof. (i) ⇔ (i′), (ii) ⇔ (ii′), (iii) ⇔ (iii′): The ⇐ implications are

clear. Now we prove (i) ⇒ (i′), the other ⇒ implications can be proved the same way.

Suppose that f : ℝ → ℝ, H ⊂ ℝ has measure zero, f is integer valued on ℝ \ H and f = f_{1} + … + f_{k} is a real valued (a_{1}, …, a_{k})-periodic decomposition of f. We need to find an almost everywhere integer valued (a_{1}, …, a_{k})-periodic decomposition of f.

By replacing H by {h+n_{1}a_{1}+…+n_{k}a_{k} : h ∈ H, n_{1}, …, n_{k} ∈ ℤ} if necessary we can suppose that the characteristic functions χ_{H} and χ_{ℝ\H} are a_{j}-periodic for every j = 1, …, k.

Applying (i) to the integer valued function χ_{ℝ\H} · f we get an almost everywhere integer valued (a_{1}, …, a_{k})-periodic decomposition χ_{ℝ\H} ·f = g_{1}+…+g_{k}. Multiplying by χ_{ℝ\H} we get

χ_{ℝ\H} · f = χ_{ℝ\H} · g_{1} + … + χ_{ℝ\H} · g_{k}.  (6)

By adding χ_{H} · f = χ_{H} · f_{1} + … χ_{H} · f_{k} to (6) we get an almost everywhere integer valued (a_{1}, …, a_{k})-periodic decomposition

f = (χ_{ℝ\H} · g_{1} + χ_{H} · f_{1}) + … + (χ_{ℝ\H} · g_{k} + χ_{H} · f_{k}).

(i) ⇒ (iv), (ii) ⇒ (iv): We prove that if (iv) is false then (i) and (ii) are also false. Suppose that \frac{1}{b_{1}} , …, \frac{1}{b_{n}} are not linearly independent over ℚ. For each b_{j} choose an a_{i_{j}} such that b_{j} is a multiple of a_{i_{j}}. Then \frac{1}{a_{i_{1}}} , …, \frac{1}{a_{in}} are also linearly dependent over ℚ, so there exists (m_{1}, …, m_{n}) ∈ ℤ×…×ℤ\{0, …, 0} such that

\frac{m_{1}}{a_{i_{1}}} + … + \frac{m_{n}}{a_{i_{n}}} = 0.  (7)

Let

f_{i_{j}}(x) = \{\frac{m_{j}}{a_{i_{j}}} x\} (j = 1, …, n),

f_{i}(x) = 0 (i ∈ {1, …, k} \ {i_{1}, …, i_{n}})

and f = f_{1} + … + f_{k}.

Then, using (7), we get

f(x) = ∑_{i=1}^{k} f_{i} = ∑_{j=1}^{n} f_{i_{j}} = ∑_{j=1}^{n} \frac{m_{j}}{a_{i_{j}}} x − ∑_{j=1}^{n} [\frac{m_{j}}{a_{i_{j}}} x] = − ∑_{j=1}^{n} [\frac{m_{j}}{a_{i_{j}}} x] ∈ ℤ.

Clearly each f_{i} is a bounded measurable a_{i}-periodic ℝ → ℝ function, so the conditions of (i) and (ii) are satisfied.

Suppose that f can be also written as f = g_{1} + … + g_{k} such that each g_{i} is an a_{i}-periodic measurable almost everywhere integer valued function. For each j = 1, …, n let h_{j} be the sum of those g_{i}-s for which b_{j} is a multiple of a_{i}. Then f = h_{1} +…+h_{n} and each h_{j} is an almost everywhere integer valued measurable b_{j}-periodic function. On the other hand f = f_{i_{1}} + … + f_{i_{n}} and each f_{i_{j}} is a measurable b_{j}-periodic function. Since b_{j}/b_{j′} ∉ ℚ for any j ≠ j′, Lemma 1.1 implies that f_{i_{j}} − h_{j} is constant almost everywhere. Since f_{i_{j}} = \{\frac{m_{j}}{a_{i_{j}}} x\}, h_{j} is almost everywhere integer valued and at least one of m_{1}, …, m_{n} is not zero, this is a contradiction.

(ii) ⇔ (iii): As we already mentioned in the Introduction, it is proved in [12] that the class of bounded measurable functions has the decomposition property; that is, a bounded measurable function f : ℝ → ℝ can be decomposed as f = f_{1} + … + f_{k} such that each f_{j} is an a_{j}-periodic real valued bounded measurable function if and only if Δ_{a_{1}} … Δ_{a_{k}}f = 0. On the other hand, by (ii), for integer valued functions the existence of a real valued bounded measurable (a_{1}, …, a_{k})-periodic decomposition is equivalent with the existence of an almost everywhere integer valued bounded measurable (a_{1}, …, a_{k})-periodic decomposition.

(iv) ⇒ (i), (iv) ⇒ (ii): First consider the case when a_{i}/a_{j} ∈ ℚ for every i, j. Then we can clearly assume that a_{j} ∈ ℤ for every j.

For t ∈ [0, 1) and n ∈ ℤ let F_{t}(n) = f(n + t) and F_{j,t}(n) = f_{j}(n + t) (j = 1, …, k). Then, applying Theorem 0.1 for each t ∈ [0, 1) for the decomposition F_{t} = F_{1,t}+…+F_{k,t} we get functions G_{j,t} : ℤ → ℤ such that F_{t} = G_{1,t}+…+G_{k,t} and each G_{j,t} is a_{j}-periodic. Letting g_{j}(n + t) = G_{j,t}(n) for each j = 1, …, k, n ∈ ℤ, t ∈ [0, 1) we get that g_{1}, …, g_{k} have all the desired properties except measurability and boundedness.

Let N be the smallest common multiple of a_{1}, …, a_{k}. For every n_{0}, n_{1}, …, n_{N−1} ∈ ℤ let

A_{n_{0},...,n_{N−1}} = {x ∈ [0, 1) : f(x) = n_{0}, f(x+1) = n_{1}, …, f(x+N −1) = n_{N−1}}.

Note that if t, t′ ∈ A_{n_{0},...,n_{N−1}} for some n_{0}, n_{1}, …, n_{N−1} ∈ ℤ then F_{t} = F_{t′}, so we may guarantee G_{j,t} = G_{j,t′} for each j in this case. Since every A_{n_{0},...,n_{N−1}} is measurable this guarantees that g_{1}, …, g_{k} are measurable.

If f_{1}, …, f_{k} are bounded, then f is also bounded, so A_{n_{0},...,n_{N−1}} is nonempty only for finitely many sequences n_{0}, n_{1}, …, n_{N−1} ∈ ℤ. Since each G_{j,t} is clearly bounded the previous paragraph guarantees that g_{1}, …, g_{k} are also bounded.

Finally we prove (iv) ⇒ (i) and (iv) ⇒ (ii) in the general case. For each equivalence class E_{i} (i = 1, …, n) of {a_{1}, …, a_{k}} with respect to the relation a ∼ b ⇔ a/b ∈ ℚ let h_{i} be the sum of those f_{j}-s for which a_{j} ∈ E_{i}. This way we get a decomposition f = h_{1} + … + h_{n} such that each h_{i} is a b_{i}-periodic measurable ℝ → ℝ function and if every f_{j} is bounded then so is every h_{i}.

Since by (iv), 1/b_{1}, …, 1/b_{n} are linearly independent over ℚ, Theorem 2.3 implies that every {h_{i}} is constant almost everywhere. By adding constants to some of the functions f_{1}, …, f_{k} we can guarantee that each {h_{i}} = 0 almost everywhere, which means that we can suppose that each h_{i} is almost everywhere integer valued. Since h_{i} = ∑_{a_{j}∈E_{i}} f_{j} and a_{j}/a_{j′} ∈ ℚ if a_{j}, a_{j′} ∈ E_{i}, the first considered case can be applied for each h_{i}. □

3 Integer valued decompositions

It is natural to ask whether we can get (everywhere) integer valued decompositions in (i), (ii) and (iii) of Theorem 2.5 or not. We will see that this depends on the answers to some questions about the general (non-measurable) case.

Question 3.1 Can one add the following statement to the list of equivalent statements of Theorem 2.5?

(i′′) If an integer valued function f : ℝ → ℤ can be decomposed as f = f_{1} + … + f_{k} such that each f_{j} is an a_{j}-periodic measurable ℝ → ℝ function then f can be also decomposed as f = g_{1} + … + g_{k} such that each g_{j} is an a_{j}-periodic integer valued measurable function.

We shall prove that this question is actually equivalent with Question 0.2. For proving the equivalence of these questions we need the following lemma, which might be known.

Lemma 3.2 For every l = 1, 2, … there exists an additive subgroup A_{l} of ℝ such that

(a) A_{l} is isomorphic to ℤ^{l} = ℤ × … × ℤ and

(b) whenever k ∈ ℕ, t_{1}, …, t_{k} ∈ A_{l} \ {0} and t_{i}/t_{j} ∉ ℚ for every i ≠ j then \frac{1}{t_{1}} , …, \frac{1}{t_{k}} are linearly independent over ℚ.

Proof. We prove by induction. For l = 1 we can choose A_{1} = ℤ.

Now we construct A_{l+1} from A_{l}. For fixed k ∈ ℕ, a = (a_{1}, …, a_{k}) ∈ A_{l}^{k}, m = (m_{1}, …, m_{k}) ∈ ℤ^{k} and n = (n_{1}, …, n_{k}) ∈ (ℤ \ {0})^{k} let

Φ_{a,m,n}(x) = \frac{n_{1}}{a_{1} + m_{1}x} + … + \frac{n_{k}}{a_{k} + m_{k}x}.

For some polynomial P_{a,m,n}(x), the function Φ_{a,m,n}(x) can be also written as

Φ_{a,m,n}(x) = \frac{P_{a,m,n}(x)}{(a_{1} + m_{1}x) · . . . · (a_{k} + m_{k}x)}.

Choose y ∈ ℝ such that y ∉ A_{l} and y is not the root of any of those polynomials P_{a,m,n}(x) that are not identically zero. This is possible since A_{l} is countable and we have only countably many polynomials and each has finitely many roots.

We claim that A_{l+1} = A_{l} + ℤy has the required properties. Since y ∉ A_{l}, A_{l+1} is indeed isomorphic to ℤ^{l+1}. Suppose that t_{1}, …, t_{k} ∈ A_{l+1} \ {0} and t_{i}/t_{j} ∉ ℚ for every i ≠ j but \frac{1}{t_{1}} , …, \frac{1}{t_{k}} are not linearly independent over ℚ. Then there exists a = (a_{1}, …, a_{k}) ∈ A_{l}^{k} and m = (m_{1}, …, m_{k}) ∈ ℤ^{k} such that t_{i} = a_{i} + m_{i}y for each i and there exists n = (n_{1}, …, n_{k}) ∈ ℤ^{k} \ {0, …, 0} such that \frac{n_{1}}{t_{1}} + … + \frac{n_{k}}{t_{k}} = 0. Then Φ_{a,m,n}(y) = 0, so P_{a,m,n}(y) = 0. We can suppose that n = (n_{1}, …, n_{k}) ∈ (ℤ \ {0})^{k} since we may simply omit every superfluous t_{j}. By the definition of y, this implies that P_{a,m,n} is identically zero, so Φ_{a,m,n}(x) = 0 for every x such that a_{i} + m_{i}x ≠ 0 for every i.

We cannot have m_{1} = … = m_{k} = 0 since then we would have t_{1}, …, t_{k} ∈ A_{l}, which would mean that A_{l} does not satisfy property (b). Let i be such that m_{i} ≠ 0. Then on one hand

lim_{x→− \frac{a_{i}}{m_{i}}} |\frac{n_{i}}{a_{i} + m_{i}x}| = ∞,

on the other hand lim_{x→− \frac{a_{i}}{m_{i}}} Φ_{a,m,n} = 0, so there must be an other term \frac{n_{j}}{a_{j}+m_{j}x} of Φ_{a,m,n} with similar property. This means that there exists a j ≠ i such that a_{j}/m_{j} = a_{i}/m_{i}. But this implies that

\frac{t_{i}}{t_{j}} = \frac{a_{i} + m_{i}x}{a_{j} + m_{j}x} = \frac{m_{i}}{m_{j}} ∈ ℚ,

which is a contradiction. □

Proposition 3.3 The answers to Question 3.1 and Question 0.2 must be the same.

Proof. First suppose that the answer is affirmative to Question 3.1 and suppose that an integer valued function f : ℝ → ℤ can be written as f = g_{1} + … + g_{k}, where each g_{j} is a real valued a_{j}-periodic function for some a_{j} ∈ ℝ. We want to show that f can be also written as f = h_{1} +…+h_{k}, where each h_{j} is an integer valued a_{j}-periodic function. It is enough to find such h_{j}-s on the additive group A generated by a_{1}, …, a_{k} since then we can define every h_{j} the same way on every coset of A. Then we can also suppose that f, g_{1}, …, g_{k} are defined also only on A, which is isomorphic to a group of the form ℤ^{l} for some l.

By Lemma 3.2 there exists an additive subgroup A_{l} of ℝ that is isomorphic to ℤ^{l} (and so to A as well) and satisfies (b) of Lemma 3.2. Hence we may assume that f, g_{1}, …, g_{k} are defined on A_{l}. For every x ∈ ℝ let

F (x) = \{ \begin{matrix}f(x) & if x ∈ A_{l}, \\ 0 & if x ∉ A_{l} \end{matrix}

and for each j = 1, …, k,

G_{j}(x) = \{ \begin{matrix}g_{j}(x) & if x ∈ A_{l}, \\ 0 & if x ∉ A_{l}. \end{matrix}

Then for each j the function G_{j} is a_{j}-periodic and also measurable since A_{j} has measure 0. Since a_{1}, …, a_{k} ∈ A_{l} and A_{l} satisfies (b) of Lemma 3.2, (iv) of Theorem 2.5 holds for a_{1}, …, a_{k}. On the other hand, affirmative answer to Question 3.1 means that (iv) of Theorem 2.5 implies (i′′) of Question 3.1. Therefore we may apply (i′′) of Question 3.1 for F = G_{1} + … + G_{k} to get a decomposition F = H_{1} + … + H_{k} such that each H_{j} : ℝ → ℤ is a_{j}-periodic. Then the restriction h_{j} of H_{j} to A_{l} (j = 1, …, k) gives a suitable decomposition of f on A.

Now we suppose that the answer to Question 0.2 is affirmative and we prove that then (i′′) of Question 3.1 is equivalent to (i), (i′), (ii), (ii′), (iii), (iii′) and (iv) of Theorem 2.5. Since in the (i) ⇒ (iv) proof of Theorem 2.5 integer valued f is constructed the same argument also proves (i′′) ⇒ (iv). So it is enough to prove (i) ⇒ (i′′)

Suppose that (i) holds, f : ℝ → ℤ, f = f_{1} + … + f_{k} and each f_{j} is a measurable a_{j}-periodic function. By (i), there exists a decomposition f = g_{1} + … + g_{k} such that each g_{j} is a measurable a_{j}-periodic almost everywhere integer valued function. All we have to do is replacing g_{j}-s by integer valued measurable functions.

Let

E_{j} = {x : g_{j}(x) ∉ ℤ} and E = (∪_{j=1}^{k}E_{j}) + a_{1}ℤ + … + a_{k}ℤ.

Then E is a set of measure zero and it is a_{j}-periodic for every j. Thus for each j the function g_{j}χ_{ℝ\E} is a_{j}-periodic. Then, by the assumption that the answer to Question 0.2 is affirmative, fχ_{E} = g_{1}χ_{E} + … + g_{k}χ_{E} implies that there exists a decomposition

fχ_{E} = F_{1} + … + F_{k}

such that each F_{j} is an integer valued a_{j}-periodic function.

For each j let

G_{j}(x) = g_{j}χ_{ℝ\E} + F_{j}χ_{E}.

Then G_{j} is clearly a_{j}-periodic and integer valued. It is also measurable since g_{j} is measurable and E is of measure zero. Since

f(x) = G_{1}(x) + … + G_{k}(x)

clearly holds both for x ∈ E and x ∈ ℝ \ E we obtained a decomposition we wanted. □

Now we prove that we cannot guarantee everywhere integer valued decompositions in (ii) and (iii) of Theorem 2.5.

Proposition 3.4 There exists a_{1}, a_{2}, a_{3} ∈ ℝ such that \frac{1}{a_{1}} , \frac{1}{a_{2}} and \frac{1}{a_{3}} are linearly independent over ℚ and a function f : ℝ → {0, 1} that has a bounded measurable real valued (a_{1}, a_{2}, a_{3})-periodic decomposition but does not have a bounded measurable integer valued (a_{1}, a_{2}, a_{3})-periodic decomposition.

Consequently one cannot replace “almost everywhere integer valued” by “integer valued” in (ii) and (iii) of Theorem 2.5.

Proof. Choose a_{1}, a_{2} and a_{3} so that a_{1} +a_{2} = a_{3} but \frac{1}{a_{1}} , \frac{1}{a_{2}} and \frac{1}{a_{3}} are linearly independent over ℚ, which is possible for example by taking a_{1}, a_{2} ∈ A_{2} such that a_{1}/a_{2} ∉ ℚ, where A_{2} is the additive subgroup of ℝ obtained by Lemma 3.2.

By Theorem 0.3 there exists a function u : ℤ × ℤ → {0, 1} that has a decomposition u = u_{1} +u_{2} +u_{3} such that u_{1}, u_{2} and u_{3} are bounded real valued periodic functions with periods (1, 0), (0, 1) and (1, 1), respectively, but u has no decomposition u = v_{1} + v_{2} + v_{3} such that v_{1}, v_{2} and v_{3} are bounded integer valued periodic functions with the same periods.

Let E = a_{1}ℤ + a_{2}ℤ,

f(x) = \{ \begin{matrix}u(n, m) & if x = na_{1} + ma_{2} (n, m ∈ ℤ), \\ 0 & if x ∉ E \end{matrix}

and for each j = 1, 2, 3,

f_{j}(x) = \{ \begin{matrix}u_{j}(n, m) & if x = na_{1} + ma_{2} (n, m ∈ ℤ), \\ 0 & if x ∉ E. \end{matrix}

Then clearly f maps to {0, 1}, f = f_{1} + f_{2} + f_{3} and each f_{j} is a_{j}-periodic, bounded and measurable (since almost everywhere zero).

But f cannot have a decomposition f = g_{1} + g_{2} + g_{3} such that each g_{j} is a_{j}-periodic, bounded and integer valued since then v_{j}(n, m) = g_{j}(na_{1} + ma_{2}) (j = 1, 2, 3) would give an integer valued bounded decomposition of u with periods a_{1}, a_{2}, a_{3}, which is impossible. □

Finally we pose two problems.

Problem 3.5 Characterize those periods a_{1}, …, a_{k} ∈ ℝ\{0} for which the existence of a bounded measurable real valued (a_{1}, …, a_{k})-periodic decomposition of an integer valued function implies the existence of a bounded measurable integer valued (a_{1}, …, a_{k})-periodic decomposition.

Theorem 2.5 implies that (iv) of Theorem 2.5 is a necessary condition but Proposition 3.4 shows that it is not sufficient. The proofs of Propositions 3.1 and 3.4 indicate that this problem must be also related to the analogue nonmeasurable problem, which seems to be also open.

Problem 3.6 Characterize those periods a_{1}, …, a_{k} ∈ ℝ \ {0} for which the existence of a bounded real valued (a_{1}, …, a_{k})-periodic decomposition of an integer valued function implies the existence of a bounded integer valued (a_{1}, …, a_{k})- periodic decomposition.

As we mentioned after Theorem 0.3, it is proved in [8] that some restriction on the periods is necessary.

References

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[4] G. H. Hardy, E. M. Wright, An introduction to the theory of numbers, Oxford, 1960.

[5] B. Farkas, T. Keleti, Sz. Gy. Révész, Invariant decomposition of functions with respect to commuting invertible transformations, submitted.

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[9] T. Keleti On the differences and sums of periodic measurable functions, Acta Math. Hung. 75 (4) (1997), 279–286.

[10] T. Keleti, Difference functions of periodic measurable functions, Fund. Math. 157 (1998), 15–32.

[11] M. Laczkovich, Sz.Gy. Révész, Periodic decompositions of continuous functions, Acta Math. Hungar. 54 (1989), no. 3-4, 329–341.

[12] M. Laczkovich, Sz. Gy. Révész, Decompositions into periodic functions belonging to a given Banach space, Acta Math. Hung. 55 (3-4) (1990), 353–363.

[13] S. Mortola, R. Peirone, The sum of periodic functions, Boll. Un. Mat. Ital. 8 2-B (1999), 393–396.

[14] T. Natkaniec, W. Wilczyński, Sums of periodic Darboux functions and measurability, Atti Sem. Mat. Fis. Univ. Modena 51 (2003), 369–376.

[15] M. Wierdl, Continuous functions that can be represented as the sum of finitely many periodic functions, Mat. Lapok 32 (1984) 107–113 (in Hungarian).

Department of Analysis Eötvös Loránd University Pázmány Péter sétány 1/c, H-1117 Budapest, Hungary

Email address: elek@cs.elte.hu
URL: http://www.cs.elte.hu/anal/keleti