arxiv: 2605.08641 · v1 · submitted 2026-05-09 · 🧮 math.DS · math.NT

Recognition: no theorem link

Invariant measure for double base expansions

Vilmos Komorni, Wenduo Huang, Yuru Zou

Pith reviewed 2026-05-12 01:26 UTC · model grok-4.3

classification 🧮 math.DS math.NT

keywords Q-expansionsgreedy maplazy mapabsolutely continuous invariant measureexact dynamical systemdouble base expansionsLebesgue measure equivalencemetric number theory

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The pith

The greedy and lazy maps on I_Q each admit a unique absolutely continuous invariant probability measure equivalent to Lebesgue measure on specified subintervals and are exact dynamical systems.

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

The paper studies Q-expansions of real numbers using a pair of bases Q = (q0, q1) satisfying q0 + q1 >= q0 q1. It introduces the greedy and lazy maps as piecewise linear transformations on the interval I_Q that generate these expansions according to their respective algorithms. The central result is that each map preserves a unique invariant probability measure that is absolutely continuous and equivalent to Lebesgue measure on a subinterval, while the full system is exact on I_Q. As a direct consequence, when the inequality is strict, Lebesgue-almost every point in I_Q possesses a continuum of distinct Q-expansions and the set of points with a unique expansion has measure zero. This describes the typical multiplicity of representations in such non-integer base systems.

Core claim

We show that the greedy and lazy maps each of which has a unique absolutely continuous invariant probability measure, equivalent to the Lebesgue measure on the intervals [0, q0/q1) and (q1/(q0(q1-1))-1, 1/(q1-1)], respectively. Furthermore, the corresponding dynamical systems are exact on I_Q. As a dynamical consequence, under the stronger condition q0 + q1 > q0 q1 the set of points having unique Q-expansions has Lebesgue measure zero, and almost every x in I_Q admits a continuum of Q-expansions.

What carries the argument

The greedy map and lazy map: the piecewise-linear maps on I_Q defined by the greedy and lazy algorithms that select the next digit in a Q-expansion.

If this is right

The invariant measures are equivalent to Lebesgue measure precisely on the stated subintervals of I_Q.
Exactness of each system implies strong mixing and ergodicity with respect to the invariant measure.
Under q0 + q1 > q0 q1, the set of points with a unique Q-expansion has Lebesgue measure zero.
Almost every x in I_Q has uncountably many distinct Q-expansions when the parameter inequality is strict.

Where Pith is reading between the lines

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

The exactness property could be used to compute the entropy of these maps or the dimension of the set of points with unique expansions.
The construction may extend to finite alphabets larger than two digits, yielding similar multiplicity results for generalized expansions.
Numerical orbit sampling under the greedy map should reproduce the density of the invariant measure on [0, q0/q1) for concrete q0, q1 values.

Load-bearing premise

The parameter restriction q0 + q1 >= q0 q1 that ensures the greedy and lazy maps are well-defined and map I_Q into itself while keeping all Q-expansions valid.

What would settle it

An explicit point or positive-measure set in I_Q whose orbit under the greedy map stays outside [0, q0/q1) or whose density with respect to Lebesgue fails to exist would contradict the claimed invariant measure.

Figures

Figures reproduced from arXiv: 2605.08641 by Vilmos Komorni, Wenduo Huang, Yuru Zou.

**Figure 1.** Figure 1: The step function hg,Q with q0 = 2.1479, q1 = 1.46557. Moreover, equality holds if and only if f or g is constant almost everywhere. Proof. See, e.g., [24] and [13]. □ Our next lemma exhibits an important property of the greedy map G (see (1.2)). Let us introduce a partial inverse H : IQ → IQ of G by the formula H(t) := 1 + t q1 . Observe that H is an increasing affine map, H(t) ≥ t for all t with equality… view at source ↗

read the original abstract

Given a pair $Q=(q_0,q_1)\in(1,\infty)^2$ with $q_0+q_1\ge q_0q_1$, a sequence $(c_i)\in\set{0,1}^\infty$ is called a $Q$-expansion of $x$ if \begin{equation*} x=\sum_{i=1}^{\infty}\frac{c_i}{q_{c_1}\cdots q_{c_i}}. \end{equation*} We primarily study the dynamical properties of the greedy and lazy maps, which are the piecewise-linear maps on the interval $I_Q=[0,\,1/(q_1-1)]$ defined by the corresponding algorithms for $Q$-expansions. We show that the greedy and lazy maps each of which has a unique absolutely continuous invariant probability measure, equivalent to the Lebesgue measure on the intervals \begin{equation*} \left[0,\frac{q_0}{q_1}\right)\qtq{and}\left(\frac{q_1}{q_0(q_1-1)}-1,\frac{1}{q_1-1}\right], \end{equation*} respectively. Furthermore, the corresponding dynamical systems are exact on $I_Q$. As a dynamical consequence, under the stronger condition $q_0+q_1>q_0q_1$ the set of points having unique $Q$-expansions has Lebesgue measure zero, and almost every $x\in I_{Q}$ admits a continuum of $Q$-expansions.

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.

Desk Editor's Note private letter to a colleague

The paper establishes unique ACIMs equivalent to Lebesgue on subintervals for the greedy and lazy maps under q0 + q1 >= q0 q1, shows exactness, and derives that almost every point has continuum many expansions when the inequality is strict.

read the letter

The core result is that for pairs Q=(q0,q1) satisfying q0 + q1 >= q0 q1, the greedy map has a unique ACIM equivalent to Lebesgue on [0, q0/q1) and the lazy map has one on the right-hand subinterval, with both systems exact on I_Q. Under the strict version of the inequality, the set of unique Q-expansions has Lebesgue measure zero and almost every x admits continuum many expansions. This follows from the maps being piecewise linear and expanding with a full branch that covers the support, so standard existence and uniqueness theorems for ACIMs apply directly, the density stays positive almost everywhere, and exactness plus the positive-length ambiguous interval forces the multiplicity result for almost all orbits. The setup ties the maps cleanly to the expansion algorithms without circularity. The work does well by making the parameter condition explicit and showing how the dynamical consequence drops out immediately once exactness is in hand. The main limitation is the restrictive condition itself, which keeps the maps inside I_Q but rules out many pairs with larger q0 or q1; the equivalence to Lebesgue also holds only on proper subintervals rather than the full interval. No load-bearing gaps appear in the logic, and the approach stays within established ergodic theory for interval maps. This is for specialists in ergodic theory or generalized base expansions who already know the single-base literature. A reader working on invariant measures for non-integer bases would find the concrete statements and the expansion consequence useful. It deserves peer review because the claims are specific, the reasoning is grounded, and the results add targeted new statements to the area.

Referee Report

0 major / 3 minor

Summary. The manuscript studies Q-expansions of real numbers in the interval I_Q = [0, 1/(q1-1)] for parameters Q = (q0, q1) satisfying q0 + q1 ≥ q0 q1. It introduces the associated greedy and lazy maps, which are piecewise-linear and expanding, and proves that each admits a unique absolutely continuous invariant probability measure equivalent to Lebesgue measure on the respective subintervals [0, q0/q1) and (q1/(q0(q1-1)) - 1, 1/(q1-1)]. The systems are shown to be exact on I_Q. Under the stricter condition q0 + q1 > q0 q1, it follows that the set of points with unique Q-expansions has Lebesgue measure zero and that almost every x ∈ I_Q admits a continuum of Q-expansions.

Significance. If the central claims hold, the paper supplies a clean dynamical-systems proof of the existence, uniqueness, and support of ACIMs for the greedy and lazy maps in this two-base setting, together with an exactness argument that directly yields the measure-zero statement for unique expansions. This fits squarely within the literature on non-integer base representations and generalized beta-transformations; the reduction to standard results for piecewise-expanding maps with a full branch is efficient and the consequence for multiplicity of expansions is a natural and falsifiable dynamical corollary.

minor comments (3)

[Abstract] Abstract, displayed equation after 'respectively': the symbol sequence 'qtq' is evidently a typesetting placeholder for 'and'; replace with the appropriate conjunction or punctuation in the final version.
[Abstract / §2] The statement that the ACIM is 'equivalent to the Lebesgue measure on the intervals' should be clarified: does this mean the invariant density is constant (hence the measure is a scalar multiple of Lebesgue restricted to the interval) or merely that the density is positive Lebesgue-almost everywhere on the support? A brief sentence distinguishing these would remove ambiguity.
[Proof of Theorem 1 / §3] When invoking 'standard results' for existence and uniqueness of ACIMs for piecewise-expanding maps, cite the precise theorem (e.g., Lasota–Yorke or its modern variants) and verify explicitly that the full-branch covering condition holds on the stated support intervals under the hypothesis q0 + q1 ≥ q0 q1.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful summary of our results on the greedy and lazy maps for Q-expansions, the positive significance assessment, and the recommendation of minor revision. No specific major comments or requested changes were identified in the report.

Circularity Check

0 steps flagged

No circularity; standard application of ergodic theory to explicitly defined maps

full rationale

The derivation begins from the explicit definition of the greedy and lazy maps on I_Q under the hypothesis q0 + q1 ≥ q0 q1, which ensures the maps send I_Q into itself and are piecewise linear with slopes q0, q1 > 1 and at least one full branch. Existence and uniqueness of the ACIMs (equivalent to Lebesgue on the stated subintervals) and exactness then follow from the standard Perron-Frobenius theory for expanding maps with a covering branch; these properties are not presupposed but derived from the forward-invariance and expansion. The measure-zero statement for unique expansions is an immediate consequence of exactness plus the positive length of the ambiguous interval. No parameters are fitted, no self-citations are load-bearing, and no result is renamed or smuggled in via prior work by the same authors. The chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the stated inequality condition on q0 and q1 together with standard background results from ergodic theory for piecewise-linear interval maps; no free parameters or new entities are introduced.

axioms (1)

domain assumption q0 + q1 >= q0 q1 ensures the greedy and lazy maps are well-defined on I_Q and the Q-expansions are valid
Explicitly required in the abstract for the maps and expansions to behave as described.

pith-pipeline@v0.9.0 · 5611 in / 1326 out tokens · 57000 ms · 2026-05-12T01:26:19.189351+00:00 · methodology

discussion (0)

Reference graph

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