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arxiv: 2504.02135 · v3 · submitted 2025-04-02 · 🧮 math.DS

Asymptotics of the Hausdorff measure for the Gauss map and its linearized analogue

Rafa{\l} Tryniecki , Mariusz Urba\'nski , Anna Zdunik This is my paper

Pith reviewed 2026-05-22 21:41 UTC · model grok-4.3

classification 🧮 math.DS
keywords Hausdorff measureGauss mapiterated function systemscontinued fractionsHausdorff dimensionasymptoticslimit setspiecewise linear
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The pith

For the first n branches of the Gauss map the h_n-dimensional Hausdorff measure H_n of the limit set satisfies (1-H_n)/((1-h_n) ln n) tending to 1 as n tends to infinity, and equivalently n(1-H_n)/ln n tending to 6/π².

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

The paper establishes asymptotic formulas for the h_n-dimensional Hausdorff measure of the limit sets arising from the first n inverse branches of the Gauss map. These sets consist of the irrationals in the unit interval whose continued-fraction expansions have entries bounded by n. The main result states that the normalized gap 1 minus the measure is asymptotically identical to the gap 1 minus the dimension, scaled by ln n. An equivalent form follows from substituting the known asymptotic for the dimension, yielding the explicit constant 6/π². The same limit relation is first proved for the piecewise-linear analogue of the system.

Core claim

The central claim is that for the IFS G_n generated by the first n inverse branches g_k(x)=1/(x+k) of the Gauss map, with limit set J_n consisting of irrationals whose continued-fraction partial quotients are at most n, one has lim (1-H_n)/((1-h_n) ln n)=1 where h_n denotes the Hausdorff dimension of J_n and H_n its h_n-dimensional Hausdorff measure; equivalently, by Hensley's dimension result, lim n(1-H_n)/ln n=6/π². The identical limit holds for the corresponding piecewise-linear IFS.

What carries the argument

The finite IFS G_n consisting of the n analytic inverse branches of the Gauss map, whose limit set J_n is the set of irrationals in [0,1] with continued-fraction entries bounded by n; the quantity whose asymptotics are derived is the h_n-dimensional Hausdorff measure H_n of this set.

If this is right

  • The same limit relation holds for the piecewise-linear analogue of the Gauss-map IFS.
  • Substituting the known asymptotic for the dimension converts the main limit into the explicit form n(1-H_n)/ln n approaching 6/π².
  • The measure H_n approaches its supremum value of 1 at a rate controlled by the dimension deficit multiplied by ln n.

Where Pith is reading between the lines

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

  • The explicit rate may be used to estimate how well finite truncations approximate the full Gauss map measure for concrete numerical purposes.
  • Analogous asymptotics could be expected for other infinite conformal IFS whose inverse branches decay like 1/k.
  • The proof method may adapt to related Diophantine approximation problems that track Hausdorff measures on Cantor sets defined by bounded partial quotients.

Load-bearing premise

The Hausdorff dimension h_n and the corresponding h_n-dimensional Hausdorff measure H_n exist, are positive and finite, for each finite truncation of the IFS.

What would settle it

Numerical evaluation for successively larger n that shows the ratio n(1-H_n)/ln n drifting away from 6/π² by a fixed positive amount would disprove the claimed limit.

Figures

Figures reproduced from arXiv: 2504.02135 by Anna Zdunik, Mariusz Urba\'nski, Rafa{\l} Tryniecki.

Figure 1
Figure 1. Figure 1: First and second generation intervals. of a single point) the left endpoint of Im coincides with right endpoint of Im−1 for all m ≤ n − 1, and I1 ∪ · · · ∪ In−1 ⊂ [0, r] If [0, r] = I1 ∪ · · · ∪In−1, then the construction ends here. Otherwise we proceed with the inductive step as follows. Since r is not the endpoint of In−1 and of any interval in Fm, m ≤ n − 1, the intervals of the collection Fn−1 do not a… view at source ↗
Figure 2
Figure 2. Figure 2: I ′ 1 compared to I1 for some α ∈ (0, 1) and all j ≥ 1. Let h ∈ (0, 1). Then the sum X∞ j=1 x h j is finite and attains its maximum for the sequence xj := (1 − α) · α j−1 , j ∈ N, and its value is equal to Sα := (1 − α) h 1 − αh . Proof. The proof is elementary and we provide it for the sake of completeness and convenience of the reader. Let Y : = ( x = (x1, x2 . . .) ∈ R N : xj ≥ 0, X∞ i=1 xi = 1, and xj+… view at source ↗
read the original abstract

Let $G(x):=\{1/x\}$ be the Gauss map. By $g_n(x)=\frac{1}{x+n}$ we denote its continuous/real analytic inverse branches. We define iterated function system (IFS) $G_n$ by limiting the collection of functions $g_k$, $k\in\mathbb N$, to the first $n$ elements, meaning that $G_n = \{g_k \}_{k=1}^n$. We are interested in the asymptotics of the Hausdorff measure of the limit set $J_n$ i. e. set consisting of irrational elements of $[0,1]$ having continued fraction expansion with entries at most $n$. In the first part of the paper, we deal with the piecewise-linear analogue of the Gauss map and resulting IFSs. We prove that \[ \lim \limits_{n \to \infty } \frac{1-H_n(J_n)}{1-h_n} \cdot \frac{1}{\ln n} = 1, \] where $J_n$ is the limit set of the piecewise-linear analogue of $G_n$, $h_n$ is its Hausdorff dimension and $H_n$ is the value of $h_n$-dimensional Hausdorff measure of the set $J_n$, $H_n:=H_{h_n}(J_n)$. In the second part, we focus on the IFS generated by the first $n$ branches of Gauss map and prove, as our main result, that $$ \lim_{n\to\infty} \frac{1-H_n}{(1-h_n)\ln n}= 1 $$ and equivalently, due to Hensley's result, $$ \lim_{n\to\infty} \frac{n(1-H_n)}{\ln n}= \frac{6}{\pi^2}, $$ where $J_n$ is the limit set of the system $G_n$, i.e. the set consisting of irrational numbers in $[0,1]$ that continued fraction expansion with entries not exceeding $n$. Similarly as for the piecewise linear map, $h_n$ is the Hausdorff dimension of $J_n$ and $H_n$ is the value of $h_n$-dimensional Hausdorff measure of the set $J_n$, $H_n:=H_{h_n}(J_n)$.

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

0 major / 2 minor

Summary. The paper studies the asymptotics of the Hausdorff dimension h_n and the h_n-dimensional Hausdorff measure H_n for the limit sets J_n of two families of finite IFS: the first n inverse branches of the Gauss map, and the corresponding piecewise-linear analogue. It proves that lim_{n→∞} (1 - H_n)/((1 - h_n) ln n) = 1 for both systems. For the Gauss IFS this is shown to be equivalent (via Hensley's theorem) to the limit lim_{n→∞} n(1 - H_n)/ln n = 6/π².

Significance. If the derivations hold, the results give a precise quantitative relation between the rate at which the Hausdorff measure approaches 1 and the rate at which the dimension approaches 1 for these truncated continued-fraction IFS. The piecewise-linear case supplies an explicit model problem, while the Gauss case directly extends Hensley's dimension asymptotics to the measure; both are of interest in fractal geometry and Diophantine approximation.

minor comments (2)
  1. [Abstract] Abstract, line 3: the notation H_n(J_n) is introduced but then H_n is used without explicit redefinition; a single consistent definition of H_n should appear once at the beginning of the abstract and again in the introduction.
  2. [Introduction] The paper invokes the open set condition to guarantee that H_n is positive and finite; a brief sentence confirming that OSC holds for both families of contractions would improve readability for readers outside IFS theory.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, the clear summary of our results, and the recommendation to accept. We have no major comments to address.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central derivation establishes the stated limit lim (1-H_n)/((1-h_n) ln n)=1 independently for the piecewise-linear IFS and for the Gauss IFS J_n, using standard IFS pressure equations and Hausdorff measure properties under the open set condition. The equivalent form lim n(1-H_n)/ln n = 6/π² is obtained by algebraic substitution of Hensley's external asymptotic result on h_n, which lies outside the paper's derivation chain and is not a self-citation. No load-bearing steps reduce by definition, by fitted inputs renamed as predictions, or by internal self-citation chains; existence of h_n and H_n follows from classical IFS theory without reference to the target asymptotics.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard properties of Hausdorff measures and dimensions for conformal IFS; no free parameters or new entities are introduced.

axioms (1)
  • standard math Existence and basic properties of Hausdorff dimension and measure for the limit sets of finite IFS
    Invoked to define h_n and H_n for each n.

pith-pipeline@v0.9.0 · 5976 in / 1180 out tokens · 54077 ms · 2026-05-22T21:41:25.163814+00:00 · methodology

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Reference graph

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