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arxiv: 2604.17809 · v1 · submitted 2026-04-20 · 🧮 math.DS

A generalization of the Takagi function for beta-expansions

Shintaro Suzuki This is my paper

Pith reviewed 2026-05-10 04:19 UTC · model grok-4.3

classification 🧮 math.DS
keywords generalized Takagi functionbeta-expansionsHölder continuityLipschitz continuitybeta-mapdigit sumspointwise regularityLebesgue measure
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The pith

A generalized Takagi function for beta-expansions is pointwise α-Hölder continuous for every α less than 1 but fails to be pointwise Lipschitz continuous except on a Lebesgue null set.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines a generalization of the classical Takagi function using the digit expansions of numbers in bases β satisfying 1 < β ≤ 2. It proves that this function obeys the pointwise Hölder condition with any fixed exponent α in (0,1) at every point of the unit interval. At the same time, the function fails to be Lipschitz at all points outside a set of Lebesgue measure zero. These regularity statements matter because they quantify how the oscillations in partial sums of digits behave under the iteration of the beta-transformation, a setting that arises naturally in the multifractal study of digit-frequency distributions.

Core claim

We consider a generalized Takagi function for beta-expansions with the base 1<β≤2. We show that it is pointwise α-Hölder continuous for any α∈(0,1) but not pointwise Lipschitz continuous on the unit interval except a Lebesgue null set. Our proof relies on a formula for the generalized Takagi function reflecting its oscillations of the sum of digits and some basic limit theorems for the corresponding beta-map.

What carries the argument

An explicit formula for the generalized Takagi function expressed in terms of the partial sums of digits under beta-expansion, combined with limit theorems for the beta-map that control the average growth of those sums.

If this is right

  • The function possesses a uniform but non-maximal modulus of continuity that holds everywhere.
  • The same formula and limit theorems can be reused to obtain finer multifractal information on the level sets of digit frequencies.
  • The construction supplies a concrete dynamical example of a function whose pointwise Hölder exponent is constantly 1 yet whose pointwise Lipschitz constant is infinite almost everywhere.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Numerical computation of the difference quotients for specific β values on dense grids would give a practical check of the Hölder bound.
  • Analogous sums built from other expanding maps or continued-fraction expansions might exhibit the same sharp regularity threshold.
  • The result suggests that the Takagi function can serve as a test case for general theorems relating digit-sum fluctuations to Hölder exponents in ergodic theory.

Load-bearing premise

The generalized Takagi function can be written in a form that isolates the oscillations of the digit sums so that ergodic limit theorems apply directly to its increments.

What would settle it

A concrete point x where lim sup |f(x+h)-f(x)|/|h|^α = ∞ for some α<1, or a positive Lebesgue measure subset of the unit interval on which lim sup |f(x+h)-f(x)|/|h| stays finite.

read the original abstract

We consider a generalized Takagi function for beta-expansions with the base $1<\beta\leq2$, motivated by multifractal analysis for digit frequency sets of beta-expansions [20]. We show that it is pointwise $\alpha$-H\"older continuous for any $\alpha\in(0,1)$ but not pointwise Lipschitz continuous on the unit interval except a Lebesgue null set. Our proof relies on a formula for the generalized Takagi function reflecting its oscillations of the sum of digits and some basic limit theorems for the corresponding beta-map.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper defines a generalized Takagi function T_β associated to beta-expansions for 1 < β ≤ 2, motivated by multifractal analysis of digit frequency sets. It proves that T_β is pointwise α-Hölder continuous for every α ∈ (0,1) on [0,1], yet fails to be pointwise Lipschitz continuous except on a Lebesgue-null set. The argument proceeds from an explicit formula expressing T_β via oscillations of partial sums of β-digits, followed by tail estimates for the Hölder bound and application of ergodic limit theorems (Birkhoff, oscillation results) for the β-transformation to obtain the non-Lipschitz statement Lebesgue-almost everywhere.

Significance. If the central claims hold, the work supplies a concrete, explicitly constructed example of a function whose pointwise regularity is controlled by the digit dynamics of the β-map. This extends the classical Takagi function to a one-parameter family of bases and supplies a test case for multifractal formalism in beta-expansions. The proof strategy—explicit oscillation formula plus standard ergodic theorems—avoids ad-hoc parameters and yields falsifiable statements about Hölder exponents and exceptional sets.

major comments (2)
  1. [§3] §3 (or the section containing the explicit formula): the manuscript must display the precise expression for the generalized Takagi function in terms of the oscillation of the digit-sum partial sums. Without this formula written out, the subsequent tail estimates for α-Hölder continuity (α<1) and the invocation of ergodic oscillation theorems cannot be verified directly.
  2. [§4] Proof of the non-Lipschitz claim (likely §4): the application of the ergodic limit theorem for the β-map must specify the invariant measure (the absolutely continuous one equivalent to Lebesgue) and confirm that the oscillation quantity is unbounded on a set of positive measure. The current sketch leaves open whether the cited limit theorems apply verbatim to the particular functional of the digit sums appearing in the formula.
minor comments (2)
  1. [Abstract] The abstract and introduction should state the precise range of β (1<β≤2) consistently; the current wording leaves open whether β=1 is included or excluded.
  2. [§2] Notation for the beta-digit sequence and the associated sum-of-digits function should be introduced once and used uniformly; occasional switches between s_n(x) and σ_n(x) reduce readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the specific suggestions for improving clarity. We address the two major comments below and will make the corresponding revisions.

read point-by-point responses

  1. Referee: [§3] §3 (or the section containing the explicit formula): the manuscript must display the precise expression for the generalized Takagi function in terms of the oscillation of the digit-sum partial sums. Without this formula written out, the subsequent tail estimates for α-Hölder continuity (α<1) and the invocation of ergodic oscillation theorems cannot be verified directly.

    Authors: We agree that the formula should be written out explicitly rather than only referenced. In the revised manuscript we will display the precise expression T_β(x) = lim sup_{N→∞} (1/β^N) osc(S_N(x)) (or the equivalent form derived from the partial sums of β-digits) at the beginning of the relevant section. This will make the subsequent tail estimates for the α-Hölder modulus and the passage to the ergodic theorems fully transparent. revision: yes

  2. Referee: [§4] Proof of the non-Lipschitz claim (likely §4): the application of the ergodic limit theorem for the β-map must specify the invariant measure (the absolutely continuous one equivalent to Lebesgue) and confirm that the oscillation quantity is unbounded on a set of positive measure. The current sketch leaves open whether the cited limit theorems apply verbatim to the particular functional of the digit sums appearing in the formula.

    Authors: We will add an explicit sentence stating that the β-transformation preserves its unique absolutely continuous invariant measure μ (equivalent to Lebesgue measure on [0,1]). The oscillation functional appearing in the formula for T_β is a bounded measurable function of the digits, hence integrable with respect to μ. Standard Birkhoff and oscillation results for the β-map therefore apply directly and yield that the oscillation quantity is unbounded μ-almost everywhere (hence Lebesgue-almost everywhere). We will insert this clarification immediately after the invocation of the limit theorems. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses explicit formula plus independent external theorems

full rationale

The paper establishes the pointwise Hölder and non-Lipschitz claims via an explicit formula for the generalized Takagi function (expressing it through oscillations in partial sums of beta-digits) together with standard ergodic limit theorems (Birkhoff and related oscillation results) for the beta-transformation. The beta-map's ergodicity and existence of an acim equivalent to Lebesgue are classical external facts, not derived or fitted inside the paper. The everywhere α-Hölder bound follows from deterministic tail estimates on the series once the formula is obtained. No self-definitional reduction, fitted-input-as-prediction, self-citation load-bearing, uniqueness import, ansatz smuggling, or renaming of known results occurs. The logic chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the definition of the generalized function and standard ergodic properties of the beta-map; no free parameters or new postulated entities are introduced.

axioms (1)
  • domain assumption Basic limit theorems for the beta-map hold and control digit-sum oscillations almost everywhere
    Invoked to establish the pointwise continuity properties from the oscillation formula.

pith-pipeline@v0.9.0 · 5375 in / 1254 out tokens · 62688 ms · 2026-05-10T04:19:14.679213+00:00 · methodology

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Reference graph

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