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arxiv: 2506.15422 · v5 · submitted 2025-06-18 · 🧮 math.NT · math.GN

Super-Dense Sets and Their Role in the Theory of Normal Numbers

Chokri Manai This is my paper

Pith reviewed 2026-05-19 09:05 UTC · model grok-4.3

classification 🧮 math.NT math.GN
keywords super-densitynormal numbersnon-normal numbersBaire propertyco-meager setscontinuous functionscomputable maps
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The pith

The set of non-normal numbers is super-dense while normal numbers are not, blocking any nowhere-constant continuous map from sending all non-normals into normals.

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

The paper introduces super-density as a topological notion of largeness on the real line. A set A is super-dense if, on any open interval and for any nowhere-constant continuous function, the image of A under that function still hits A. It proves that among sets possessing the Baire property this notion coincides exactly with being co-meager. Applying the characterization to the classical sets of normal and non-normal numbers shows that non-normal numbers are super-dense while normal numbers are not. The distinction immediately rules out the existence of any nowhere-constant continuous function that would carry every non-normal number into the normal numbers, yet the paper also supplies an explicit computable function that does the opposite and a general construction that preserves non-normality under any countable collection of such maps.

Core claim

The central claim is that a subset A of the reals is super-dense precisely when, for every nonempty open interval I and every nowhere-constant continuous map φ defined on I, the set φ(I ∩ A) intersects A. Sets with the Baire property are super-dense if and only if they are co-meager. Consequently the set of non-normal numbers is super-dense, while the set of normal numbers is not; in particular no nowhere-constant continuous function can send the entire set of non-normal numbers into the set of normal numbers.

What carries the argument

Super-density: the intersection property that A meets φ(I ∩ A) for every open interval I and every nowhere-constant continuous φ on I.

If this is right

  • No nowhere-constant continuous function can map every non-normal number to a normal number.
  • There exists a computable nowhere-constant continuous function that maps every normal number to a non-normal number.
  • For any countable collection of nowhere-constant continuous functions there is a non-normal number x whose images under all of them remain non-normal.
  • There exists a non-normal real x such that e raised to αx is non-normal for every nonzero algebraic number α.

Where Pith is reading between the lines

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

  • The same super-density argument could be tested on other Diophantine or arithmetic properties that are known to be meager yet full measure.
  • The explicit construction of a map sending normals to non-normals might be adapted to produce numbers that avoid normality in several bases simultaneously.
  • One could ask whether super-density is preserved under countable intersections or under addition of a fixed constant.

Load-bearing premise

The set of normal numbers contains no co-meager subset that possesses the Baire property.

What would settle it

An explicit nowhere-constant continuous function φ together with an interval I such that φ maps every point of I ∩ (non-normal numbers) into the normal numbers.

read the original abstract

We introduce and study a new topological notion of the size for subsets of the real line, called \emph{super-density}. A set $A\subset\mathbb{R}$ is super-dense if for every non-empty open interval $I$ and every nowhere constant continuous function $\varphi\colon I\to\mathbb{R}$, we have $\varphi(I\cap A)\cap A\neq\emptyset$. We first establish basic properties of super-dense sets. Our main topological result characterizes them within the framework of Baire category: a set with the Baire property is super-dense if and only if it is co-meager. We then investigate the implications for the theory of normal numbers. We prove that the set of non-normal numbers is super-dense, whereas the set of normal numbers is not. Consequently, no nowhere constant continuous function can map all non-normal numbers to normal numbers. Conversely, we explicitly construct a computable nowhere constant continuous function that maps all normal numbers to non-normal numbers. Finally, we provide a constructive algorithm that, given any countable family of nowhere constant continuous functions, produces a real number $x$ such that $x$ and all its images under these functions are non-normal. As a corollary, we obtain the existence of a non-normal number $x$ such that $e^{\alpha x}$ is non-normal for every non-zero algebraic $\alpha$.

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 introduces super-dense sets A ⊂ ℝ, defined so that for every nonempty open interval I and every nowhere-constant continuous φ: I → ℝ one has φ(I ∩ A) ∩ A ≠ ∅. It proves that a set with the Baire property is super-dense if and only if it is co-meager, then claims that the non-normal numbers are super-dense while the normal numbers are not. From this it deduces that no nowhere-constant continuous map sends all non-normal numbers into the normal numbers, constructs an explicit computable counter-example map in the opposite direction, and supplies a diagonalization algorithm that, given any countable family of such maps, produces a non-normal x whose images under the family are also non-normal; a corollary asserts the existence of a non-normal x such that e^{αx} is non-normal for every nonzero algebraic α.

Significance. If the central claims hold, the new notion supplies a Baire-category characterization of a strong form of topological largeness and yields concrete mapping and existence results for normal numbers that are constructive. The explicit computable function and the countable-family algorithm are genuine strengths that could be useful for further work on invariance properties of normal numbers.

major comments (2)
  1. [§3] §3 (Baire-category status of normal numbers): the argument that non-normal numbers are co-meager (hence super-dense) while normal numbers are meager rests on showing that the latter set contains no co-meager subset with the Baire property. This directly contradicts the standard result that the normal numbers form a comeager G_δ set. Because the paper’s Theorem 2.1 equates super-density with co-meagerness for BP sets, the reversal undermines the mapping claims in §4 and the algorithm in §5. A concrete test is to verify whether the countable union over all blocks and all rational deviations from the correct frequency is shown to be meager or, conversely, whether the intersection defining normality is shown to be dense G_δ.
  2. [§4] §4, first displayed claim after Theorem 2.1: the statement that “no nowhere-constant continuous function can map all non-normal numbers to normal numbers” is derived from the super-density of the non-normal set. If the Baire-category identification in §3 is reversed, this implication fails and the subsequent explicit construction in the opposite direction becomes the only surviving direction.
minor comments (2)
  1. [§2] The definition of super-density would be clearer if an explicit example of a nowhere-constant continuous φ and a set A were given immediately after the definition.
  2. A short paragraph recalling the known Baire-category and measure-theoretic status of normal numbers (with one or two standard references) would help readers assess the novelty of the application.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting a fundamental inconsistency between our claims in §3 and well-established results on the Baire category of normal numbers. We agree that a major revision is necessary and address each point below.

read point-by-point responses

  1. Referee: [§3] §3 (Baire-category status of normal numbers): the argument that non-normal numbers are co-meager (hence super-dense) while normal numbers are meager rests on showing that the latter set contains no co-meager subset with the Baire property. This directly contradicts the standard result that the normal numbers form a comeager G_δ set. Because the paper’s Theorem 2.1 equates super-density with co-meagerness for BP sets, the reversal undermines the mapping claims in §4 and the algorithm in §5. A concrete test is to verify whether the countable union over all blocks and all rational deviations from the correct frequency is shown to be meager or, conversely, whether the intersection defining normality is shown to be dense G_δ.

    Authors: We fully acknowledge that our argument in §3 is incorrect. The normal numbers do form a comeager G_δ set, so the non-normal numbers are meager. Our attempt to show that the non-normal set is super-dense (and therefore co-meager among BP sets) contains an error in the assessment of the relevant countable union. We will revise §3 to remove the erroneous claim that non-normal numbers are super-dense and to confirm, as the referee suggests, that the intersection over all blocks and rational frequencies that defines normality is a dense G_δ set. This correction will be accompanied by a brief reference to the standard literature establishing the comeager status of normal numbers. revision: yes

  2. Referee: [§4] §4, first displayed claim after Theorem 2.1: the statement that “no nowhere-constant continuous function can map all non-normal numbers to normal numbers” is derived from the super-density of the non-normal set. If the Baire-category identification in §3 is reversed, this implication fails and the subsequent explicit construction in the opposite direction becomes the only surviving direction.

    Authors: We agree that the derivation of the claim in §4 relies on the now-retracted super-density of the non-normal numbers and therefore cannot be maintained. We will delete this unsupported statement. The explicit computable nowhere-constant continuous map constructed later in §4, which sends normal numbers to non-normal numbers, does not depend on the Baire-category argument and will be retained with clarified wording indicating its independent status. We will also review the algorithm in §5 to determine whether any part of it can be salvaged without the super-density hypothesis; if not, that section will be removed or substantially rewritten. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines a new notion of super-density and proves via standard Baire-category methods that a Baire-property set is super-dense precisely when co-meager. It then applies this characterization to the known category status of normal and non-normal numbers, followed by explicit constructions of continuous functions and a diagonalization algorithm. None of these steps reduces a claimed result to a quantity defined in terms of the paper's own fitted parameters or to a self-citation chain; the central claims remain independent of the inputs they are derived from.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper introduces a new definition and relies on standard facts from Baire category theory and the definition of normal numbers. No free parameters are fitted to data. No new physical or mathematical entities are postulated beyond the defined super-dense sets.

axioms (1)
  • standard math Baire category theorem and basic properties of sets with the Baire property
    Invoked to characterize super-dense sets as co-meager sets.

pith-pipeline@v0.9.0 · 5768 in / 1462 out tokens · 36468 ms · 2026-05-19T09:05:46.573984+00:00 · methodology

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Reference graph

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