Paper Skeleton: The Incommensurable Decomposition Problem

Working title: Incommensurable periodic decompositions: constructions, obstructions, and the limits of measure

Author: Alonso Espinosa Domínguez

Date: March 29, 2026 — Working outline

Abstract (sketch)

We introduce the incommensurable decomposition problem (IDP): given a periodic function \(h\), when does \(h = f + g\) with \(P_f \cap P_g = \{0\}\)? This is a strengthening of the classical periodic decomposition problem (PDP). We construct AC-dependent examples of incommensurable decompositions where \(P_{f+g} \not\subseteq P_f \oplus P_g\), and discuss obstructions to extending these constructions. We investigate how far the Cage of Continuity (Theorem 2.1) can be strengthened, reducing the problem in the measurable case to a topological question about points of continuity, and discuss the role of the Axiom of Choice.

Section 1. Introduction

Goal: Motivate the problem, state main results informally, outline paper structure.

Origin: Axler’s exercise — is \(\mathscr{P}\) a subspace of \(\mathbb{R}^\mathbb{R}\)?
Quick summary: the answer is no (continuous case), but the failure is not universal.
The classical PDP (Farkas-Révész): \(h = f + g\) with \(f\) \(p\)-periodic, \(g\) \(q\)-periodic iff \(\Delta_p \Delta_q h = 0\).
The IDP as a new problem: same setup, but require \(P_f \cap P_g = \{0\}\).
Why the distinction matters: the PDP construction allows inherited periods; the IDP forbids them. In particular, if \(P_h\) has periods outside \(p\mathbb{Z} \oplus q\mathbb{Z}\), the PDP construction produces \(f, g\) that inherit those periods, so \(P_f \cap P_g \neq \{0\}\).
Overview of results.

Status: To be written last. Depends on which results make the final cut.

Section 2. Preliminaries

Goal: Fix notation, state known results.

2.1 Period modules and basic properties

Definition of \(P_f\), strictly periodic, rank.
Difference operators \(\Delta_q\), linearity, commutativity.
\(P_f\) dense iff \(f\) not strictly periodic.
Dense period module implies constant or nowhere continuous (Proposition 1.2).

2.2 Commensurability and incommensurability

Definitions.
Incommensurable functions with incommensurable periods are nowhere continuous.

2.3 The Cage of Continuity (Theorem 2.1 from blog)

If \(f, g\) continuous periodic, then \(f + g\) periodic implies commensurability.
Proof sketch (via \(\Delta_q f\) having dense period module).

2.4 Points of continuity and difference operators

\(\mathcal{C}(f)\) denotes the set of points of continuity of \(f\).
\(\mathcal{C}(f)\) is always a \(G_\delta\) set.
Key subtlety: \(\mathcal{C}(f) \neq \emptyset\) does not imply \(\mathcal{C}(\Delta_q f) \neq \emptyset\). A function can have points of continuity even when all its nontrivial difference operators are nowhere continuous (Exercise 2 from blog).
The question of whether \(\mathcal{C}(f) \cap (\mathcal{C}(f) + nq) = \emptyset\) is possible for all \(n \in \mathbb{Z} \setminus \{0\}\) with \(\mathcal{C}(f)\) dense remains open.

2.5 Mirotin’s results

Theorem C (Mirotin): If \(f\) is a.e. periodic, measurable, and not a.e. constant on a set of positive measure, then any two periods of \(f\) are commensurable (i.e., \(f\) is strictly periodic).
Theorem 2 (Mirotin): If \(f_1, \ldots, f_n\) are a.e. \(T_i\)-periodic, measurable, not a.e. constant, and all subsums of fewer than \(n\) summands are not a.e. constant, then \(f_1 + \cdots + f_n\) is a.e. periodic iff the \(T_i\) are commensurable.
Consequence for \(n = 2\): measurable, not a.e. constant, incommensurable periodic functions cannot have an a.e. periodic sum. Period.

2.6 Connection to the classical PDP

Statement of the PDP characterization: \(\Delta_p \Delta_q h = 0\) is necessary and sufficient (with AC).
How the PDP construction works (Farkas-Révész 2013).
Key observation: the PDP construction can produce \(f, g\) with \(P_f \cap P_g \neq \{0\}\) when \(P_h\) has periods outside \(p\mathbb{Z} \oplus q\mathbb{Z}\). The IDP asks whether an alternative decomposition with \(P_f \cap P_g = \{0\}\) exists.

Status: Mostly written (blog posts 1–2 + context document). Needs translation to paper format and integration of Mirotin.

Section 3. The Nowhere Bounded Theorem and its consequences

Goal: Present the chain of results from difference operators to nowhere-boundedness.

3.1 Newton’s Forward Difference Formula

Statement via simplicial numbers.
When \(\Delta_q^N f = 0\): \(f(x + nq)\) is polynomial in \(n\).

3.2 The Commensurability Forcing Lemma

If \(q \notin P_f\) and \(\Delta_q^2 f = 0\), then \(q \notin \operatorname{span}_\mathbb{Q}(P_f)\).

3.3 The Nowhere Bounded Theorem

If \(f\) periodic, \(q \notin P_f\), \(\Delta_q^N f = 0\): \(f\) is nowhere bounded.
Corollary: \(\mathcal{C}(f) = \emptyset\).

3.4 The Reduction Lemma

If \(\Delta_r \Delta_q f = 0\) and \(r \in \operatorname{span}_\mathbb{Q}(P_f \oplus q\mathbb{Z})\): reduces to \(\Delta_{q'}^2 f = 0\).
Consequence: in an incommensurable decomposition with \(r\) in the \(\mathbb{Q}\)-span, \(f\) is nowhere bounded.

Status: Fully developed in previous chats. Ready to write up.

Section 4. Elementary constructions

Goal: Exhibit concrete incommensurable decompositions.

4.1 Incommensurable functions with periodic sum

Construction: \(G = p\mathbb{Z} + q\mathbb{Z}\), \(f(mp+nq) = n\), \(g(mp+nq) = -m\), zero outside \(G\).
Result: \(P_f = p\mathbb{Z}\), \(P_g = q\mathbb{Z}\), \(P_{f+g} = (p+q)\mathbb{Z}\).
Functions are measurable, zero a.e., strictly periodic, incommensurable.
Note: here \(P_{f+g} = (p+q)\mathbb{Z} \subseteq P_f \oplus P_g\), so the sum has no “unexpected” periods.

4.2 Hidden periods and the difficulty of incommensurability

Construction: \(G = p\mathbb{Z} \oplus q\mathbb{Z} \oplus r\mathbb{Z}\) with \(p, q, r\) \(\mathbb{Q}\)-independent.
\(f(ap+bq+cr) = b+c\), \(g(ap+bq+cr) = a-c\), zero outside \(G\).
Period modules: \(P_f = p\mathbb{Z} \oplus (q-r)\mathbb{Z}\), \(P_g = q\mathbb{Z} \oplus (p+r)\mathbb{Z}\).
\(P_f \cap P_g = (q-p-r)\mathbb{Z} \neq \{0\}\) — functions are commensurable, not incommensurable.
Purpose of this example: not as a result in itself (commensurable functions trivially have \(\operatorname{span}_\mathbb{R}(\{f,g\}) \subseteq \mathscr{P}\)), but as an illustration of a key phenomenon: a naive attempt to build incommensurable generators for a subspace of \(\mathscr{P}\) collapses into commensurability via hidden periods. The period modules expand (from \(p\mathbb{Z}\) to \(p\mathbb{Z} \oplus (q-r)\mathbb{Z}\)) and absorb what should have been “outside” periods.
The identity \(\beta(-1) + \alpha(1) + (\alpha - \beta)(-1) = 0\) as the algebraic mechanism producing the universal shared period.

4.3 The difficulty of \(P_h \not\subseteq P_f \oplus P_g\)

In §4.1, \(P_{f+g} \subseteq P_f \oplus P_g\).
In §4.2, although \(r\) appears to be outside \(p\mathbb{Z} \oplus q\mathbb{Z}\), the true period modules are larger: \(r = q - (q-r)\) with \(q - r \in P_f\) and \(q \in P_g\), so \(r \in P_f \oplus P_g\).
Achieving \(P_h \not\subseteq P_f \oplus P_g\) genuinely requires preventing this “absorption.” This is what the Hamel basis construction (§5) accomplishes.

Status: Constructions verified, proofs complete. Framing clarified.

Section 5. Constructions via the Axiom of Choice

Goal: Produce genuinely incommensurable examples with unexpected period behavior.

5.1 The Hamel basis construction

Construction: \(G = \operatorname{span}_\mathbb{Z}(B)\), component projections \(\pi_y\), functions \(f = \pi_{\phi_1(x+G)}\), \(g = -\pi_{\phi_2(x+G)}\).
Choice function \(c: \mathbb{R}/G \to \mathbb{R}\) normalized with \(c(G) = 0\).
Result: \(P_f = p\mathbb{Z}\), \(P_g = q\mathbb{Z}\) (full proof via case analysis), \(f\) and \(g\) genuinely incommensurable.
\(P_{f+g} \supseteq \operatorname{span}_\mathbb{Z}((B \setminus \{p,q\}) \cup \{p+q\})\) — periods outside \(P_f \oplus P_g = p\mathbb{Z} \oplus q\mathbb{Z}\).
This is the only confirmed construction where \(P_{f+g} \not\subseteq P_f \oplus P_g\) with \(P_f\) and \(P_g\) exactly determined.
This shows that the constraint from the Reduction Lemma (§3.4) can be violated with AC.

5.2 Extension of elementary constructions to all cosets

The construction in §4.1 can be extended via a choice function on \(\mathbb{R}/G\) to produce functions that are not a.e. constant (and not measurable).
Similarly for §4.2. Period modules and algebraic properties carry over.
These extensions require AC (essentially Vitali-type constructions).

5.3 Discussion

The Hamel basis construction gives incommensurable \(f, g\) with periodic sum with \(P_{f+g} \not\subseteq P_f \oplus P_g\), but does NOT give \(\operatorname{span}_\mathbb{R}(\{f,g\}) \subseteq \mathscr{P}\).
The elementary constructions (§4) achieve \(\operatorname{span}_\mathbb{R}\) but only with commensurable functions (which is trivial).
This gap motivates the conjecture in §6.

Status: Construction verified, \(P_f = p\mathbb{Z}\) proof complete. Relationship to elementary constructions clarified.

Section 6. Strengthening the Cage of Continuity

Goal: Investigate how far Theorem 2.1 (Cage of Continuity) can be pushed beyond continuous functions.

6.1 The Strong Theorem 2.1

Conjecture (Strong Cage of Continuity). If \(f, g\) are periodic functions with \(\mathcal{C}(f) \neq \emptyset\) and \(\mathcal{C}(g) \neq \emptyset\), then \(f + g\) periodic implies \(f, g\) commensurable.

This would be optimal: without any continuity assumption, incommensurable sums exist (§4.1); with full continuity, commensurability is forced (Theorem 2.1). The question is whether a single point of continuity suffices.

6.2 Decomposition into cases

By Mirotin’s Theorem 2, we can separate the problem:

Case A: \(f\) or \(g\) not a.e. constant (and measurable). Mirotin’s Theorem 2 already implies commensurability — a single point of continuity is more than enough. (In fact, measurability + not a.e. constant suffices without any continuity assumption.)

Case B: \(f\) and \(g\) both a.e. constant (the “measure-invisible” case). This is the hard case. Here \(f, g\) are a.e. zero (WLOG), measurable, with \(\mathcal{C}(f) \neq \emptyset\) and \(\mathcal{C}(g) \neq \emptyset\). The measure-theoretic tools (Mirotin, Steinhaus) lose resolution — the functions are trivial from the perspective of Lebesgue measure, but their algebraic and topological structure is rich.

Case C: \(f\) or \(g\) not measurable. Requires AC. The problem is open here; it is not clear whether Mirotin-style arguments can be replaced by purely algebraic/topological ones.

6.3 Analysis of Case B

If \(f\) is a.e. zero and \(\mathcal{C}(f) \neq \emptyset\), then \(f(x_0) = 0\) at any point of continuity \(x_0\), and \(f = 0\) on an open neighborhood of \(x_0\). If \(f\) is \(p\)-periodic, then \(f = 0\) on the open set \(U + p\mathbb{Z}\), which is dense. So \(f\) vanishes on an open dense set; its support (where \(f \neq 0\)) is a closed, nowhere dense set of measure zero — a “dust.”

The question becomes: can the sum of two “periodic dusts” with incommensurable periods be periodic?

This reduces to a topological question about \(\mathcal{C}(f)\):

If \(q \notin P_f\), then \(\Delta_q f\) is periodic (with periods in \(P_f\)).
For the Cage of Continuity argument to work, we need \(\mathcal{C}(\Delta_q f) \neq \emptyset\).
\(\mathcal{C}(\Delta_q f) \supseteq \mathcal{C}(f) \cap (\mathcal{C}(f) - q)\), so it suffices to have \(\mathcal{C}(f) \cap (\mathcal{C}(f) + q) \neq \emptyset\) for some \(q \in P_g \setminus \{0\}\).
But \(\mathcal{C}(f)\) can avoid all its \(q\)-translates. This is the key obstacle: even though \(\mathcal{C}(f)\) is \(G_\delta\) and dense (and \(p\)-periodic), it is possible in principle that \(\mathcal{C}(f) \cap (\mathcal{C}(f) + nq) = \emptyset\) for all \(n \in \mathbb{Z} \setminus \{0\}\), since \(q\mathbb{Z}\) is discrete and a dense \(G_\delta\) of measure zero can avoid a discrete set of translates.

6.4 The topological question

Open Question. Let \(A \subseteq \mathbb{R}\) be \(G_\delta\), dense, and \(p\)-periodic (i.e., \(A + p = A\)). Let \(q\) with \(p/q \notin \mathbb{Q}\). Must \(A \cap (A + nq) \neq \emptyset\) for some \(n \in \mathbb{Z} \setminus \{0\}\)?

If yes: Strong Theorem 2.1 holds in Case B, and combining with Mirotin (Case A), the Strong Cage of Continuity holds for all measurable functions.

If no: there exist a.e.-constant, measurable, incommensurable periodic functions with points of continuity whose sum can (potentially) be periodic. The Strong Theorem 2.1 would fail, and the “weakest cage” would be strictly between \(\mathcal{C}(f) \neq \emptyset\) and full continuity.

6.5 Remarks

The analogous question for full measure instead of dense \(G_\delta\) is trivially yes (by Steinhaus). The difficulty is specific to topologically large but metrically small sets.
For \(\mathcal{C}(f)\) dense but not \(p\)-periodic, the answer is likely no: a dense \(G_\delta\) of measure zero can avoid a discrete set of translates. The \(p\)-periodicity of \(\mathcal{C}(f)\) (inherited from \(f\)) is the additional structure that might force intersection.

Status: Problem cleanly reduced. Topological question formulated. Answer unknown.

Section 7. The commensurability conjecture for \(\operatorname{span}_\mathbb{R}\)

Goal: State and discuss the conjecture on real spans.

7.1 Statement

Conjecture. If \(f, g\) are periodic functions with \(\operatorname{span}_\mathbb{R}(\{f,g\}) \subseteq \mathscr{P}\), then \(f\) and \(g\) are commensurable.

Note: for commensurable \(f, g\), \(\operatorname{span}_\mathbb{R}(\{f,g\}) \subseteq \mathscr{P}\) is trivially true. The content of the conjecture is that no other case is possible.

7.2 Evidence

All known constructions achieving \(\operatorname{span}_\mathbb{R}(\{f,g\}) \subseteq \mathscr{P}\) produce commensurable functions (§4.2 and its AC extension).
The construction in §4.2 shows that naive attempts to build incommensurable generators collapse into commensurability.
Structural constraint: for each \(\alpha\), \(\alpha f + g\) has some period \(s_\alpha\). The set \(S = \{s_\alpha\}_{\alpha \in \mathbb{R}}\) satisfies \(\Delta_{s_\alpha}\Delta_{s_\beta} f = 0\) for all \(\alpha \neq \beta\), forcing \(P_{\Delta_{s_\beta} f} \supseteq S\). This means every \(\Delta_{s_\beta} f\) has a period module containing a set of cardinality \(\mathfrak{c}\), and hence its period module is dense (even uncountably generated). By Proposition 1.2, each \(\Delta_{s_\beta} f\) is either constant or nowhere continuous. These are severe constraints on \(f\).

7.3 Partial results and approaches

The cardinality argument (incomplete): each \(s\) with \(\Delta_s f \neq 0\) serves at most one \(\alpha\), but uncountably many singletons can cover \(\mathbb{R}\). Does not close.
Mirotin route (measurable case): For measurable \(f, g\) not a.e. constant, Mirotin’s Theorem 2 already implies commensurability (even for \(f + g\) alone, without the full \(\mathbb{R}\)-span). For a.e. constant \(f, g\), \(\Delta_{s_\beta} f\) is a.e. zero for every \(\beta\), so the structural constraints become a.e. trivial. The conjecture in the measurable case reduces to the same topological question as the Strong Theorem 2.1 (§6.4).
The transfinite approach (speculative): attempting to construct \(\operatorname{span}_\mathbb{R}(\{f,g\}) \subseteq \mathscr{P}\) with incommensurables via transfinite recursion, each step “purchases” periodicity for one new \(\alpha\) but requires growing period modules. The accumulation of periods may not converge, suggesting impossibility. Could potentially be reversed into a proof by contradiction.

7.4 Consequences if true

Would imply that \(|F| \leq 1\) for mutually incommensurable \(F\) with \(\operatorname{span}_\mathbb{R}(F) \subseteq \mathscr{P}\).
Would establish “hidden commensurability” as a necessary feature of subspaces of \(\mathscr{P}\).
Would provide a new type of rigidity theorem for periodic functions, complementing the Cage of Continuity.

Status: Conjecture formulated, evidence assembled, proof incomplete. Intimately connected to §6.

Section 8. Axiomatics and measurability

Goal: Discuss what requires AC, what doesn’t, and connections to set-theoretic principles.

8.1 What requires AC and what doesn’t

Elementary constructions (§4): no AC needed. Measurable, a.e. zero.
Coset extensions (§5.2): require a choice function on \(\mathbb{R}/G\), essentially a Vitali-type construction.
Hamel basis construction (§5.1): requires a Hamel basis, which is at least as strong as a Vitali set.

8.2 The hierarchy of choice principles

Vitali sets: existence is strictly weaker than AC and BPI.
Hamel bases for \(\mathbb{R}/\mathbb{Q}\): imply Vitali sets, but reversal probably false.
ZF+DC is consistent with all sets measurable (Solovay 1970, assuming inaccessible cardinal).
Key references: Howard-Rubin, Herrlich, Blass 1984, Schindler.

8.3 Measure-theoretic invisibility

The elementary constructions (§4) are measurable, a.e. constant, and trivial from the perspective of Lebesgue measure. But they have exact period modules, hidden shared periods, and rich algebraic structure that measure theory cannot detect.
Mirotin’s results characterize the measurable, non-a.e.-constant case completely. The a.e.-constant case is where measure theory loses resolution and algebraic/topological tools are needed.
This motivates the approach in §6: studying a.e.-constant functions via their points of continuity rather than their measure-theoretic properties.

8.4 Open questions

Are the non-measurable constructions in this paper (§5) equivalent (over ZF) to the existence of Vitali sets, or do they require strictly more/less choice?
Is the existence of incommensurable periodic functions whose sum is periodic with \(P_{f+g} \not\subseteq P_f \oplus P_g\) equivalent to some specific choice principle?

Status: Conceptual framework established. Requires careful literature verification for the set-theoretic claims.

Section 9. Open problems

Goal: Collect remaining open questions for future work.

The Strong Cage of Continuity (§6.1): does \(\mathcal{C}(f) \neq \emptyset\) and \(\mathcal{C}(g) \neq \emptyset\) with \(f + g\) periodic force commensurability?
The topological question (§6.4): for \(A\) a \(G_\delta\), dense, \(p\)-periodic subset of \(\mathbb{R}\), and \(q\) with \(p/q \notin \mathbb{Q}\), must \(A \cap (A + nq) \neq \emptyset\) for some \(n \neq 0\)?
The \(\operatorname{span}_\mathbb{R}\) conjecture (§7.1): does \(\operatorname{span}_\mathbb{R}(\{f,g\}) \subseteq \mathscr{P}\) force commensurability?
Maximal dimension of subspaces of \(\mathscr{P}\): what is the largest dimension of a subspace \(V \subseteq \mathscr{P}\)? (Note: \(V\) always contains commensurable functions; the question is about the “incommensurability content” of \(V\).)
Dense points of continuity: can \(\mathcal{C}(f)\) be dense for non-strictly-periodic \(f\) when \(\Delta_q f\) is nowhere continuous for all \(q \notin P_f\)?
Axiom-of-Choice equivalences: precise placement of the constructions in §5 in the choice hierarchy.
Weakest regularity forcing commensurability: what is the largest family \(\mathscr{F} \supset \mathscr{C}\) such that \(f, g \in \mathscr{F}\) periodic with \(f + g\) periodic implies commensurability? (The “strongest cage.”)

Dependencies and writing order

§2 first (prerequisites, mostly done).
§3 next (Nowhere Bounded theorem, fully developed).
§4 next (elementary constructions, verified).
§5 next (AC constructions, verified).
§6 depends on progress with the conjecture.
§7 can be written independently.
§1 and §8 last (introduction and open problems).

Target venue (tentative)

If commensurability conjecture resolved: Real Analysis Exchange or Proceedings of the AMS.
If conjecture remains open: American Mathematical Monthly or Mathematics Magazine (as a problem-posing/survey article with novel constructions).