bibkey: jelinek2025gowers authors: Pascal Jelinek year: 2025 title: Gowers norms for linearly recurrent numeration systems doi: 10.48550/arXiv.2510.16947 url: https://arxiv.org/abs/2510.16947v1 claim: “Remark 1.5 supplies a Rauzy-fractal/torus direction for digital observations. The general decay statement in Theorem 1.4 has unresolved resonance and truncation discrepancies in v1; it is not admitted here as a weighted FIB or Robin estimate.” strata_touched: [] license: citation-only triage: anchor

Recursive digital geometry and the missing arithmetic transport

The inspected source is arXiv:2510.16947v1, submitted 2025-10-19. Its author TeX was also inspected. The arXiv abstract and API identify v1 as the latest version in the retrieval on 2026-10-05 UTC. This is a version check of this paper, not a claim to have surveyed every recent result. The source check covers Definition 1.2, Theorem 1.4, Remark 1.5, Lemmas 2.1–2.2, Proposition 3.1 and the final displayed estimate in the proof of Theorem 1.4; the complete proof has not been independently certified or Lean-verified.

What the geometric direction preserves

Definition 1.2 requires the finiteness property, Property F, of the associated beta-expansion. Being Pisot alone does not discharge this condition. Lemmas 2.1–2.2 use this property to separate digit blocks by sufficient zero buffers before addition. The buffers depend on the addition problem; the two-state rule forbidding adjacent Zeckendorf ones is not by itself a uniform carry bound for every arithmetic operation.

Remark 1.5 explicitly distinguishes the displayed digital cube average from a genuine Gowers norm in most noninteger-base systems. It outlines an observation on a Rauzy fractal, passage to a torus using a tiling, and integration against a suitable measure. The remark does not specify a ready-to-use weighted norm inequality for the project’s arithmetic input. An interval image of legal addresses does not, by itself, supply that tiling, group operation, measure, or transport of arithmetic weights.

For the project’s five-mode address, the digit-count observable is

$$\nu (null)=0,\nu (2)=\nu (3)=\nu (5)=1,\nu (25)=2.$$

Use $G_i=F_{i+2}$, so $G_0=1$, and retain the unit digit separately. The first $L$ windows occupy positions $1,\dots ,3L$. With the inclusive cutoff in the paper’s equation (1.1), their truncated digit count at $\lambda =3L$ is the unit digit plus the sum of these window counts. This parameter identification concerns a digit phase. The paper’s dominant root, also called $\beta$, is distinct from the Binet error coefficients ${\beta }_d$ in FIB §384.

The displayed general phase estimate cannot be imported unchanged

Theorem 1.4, equation (1.3), prints decay with $\Vert \theta {\Vert }^2$ in the exponent. Proposition 3.1 and the last displayed line of the proof instead contain $\Vert h\theta {\Vert }^2$. The source prints $h=a_0-1+a_1+\cdots +a_d$ although Definition 1.1 names the recurrence coefficients $a_1,\dots ,a_d$. These are discrepancies in the original TeX, not only in PDF extraction. No corrected general theorem is asserted here.

The classical integer-base resonance is a necessary source check. In base three, $s_3(n)\equiv n(\mathrm{mod}2)$. Take $s=2$, $\theta =1/2$, $G_{\lambda }=3^{\lambda }$, and the paper’s inclusive truncation $s_{3,\lambda }(n)={\sum }_{i=0}^{\lambda }{\delta }_i(n)$. Every cube vertex for $0\le n_0,n_1,n_2<3^{\lambda }$ is below $3^{\lambda +1}$, so this truncation retains all its digits. Consequently the four phases multiply to one:

$$\frac{1}{3^{3\lambda }}\sum _{n_0,n_1,n_2<3^{\lambda }}(-1{)}^{s_{3,\lambda }(n_0)-s_{3,\lambda }(n_0+n_1)-s_{3,\lambda }(n_0+n_2)+s_{3,\lambda }(n_0+n_1+n_2)}=1.$$

This tests equation (1.3) with its cube dimension read as $s$; the display also uses a different letter $k$ in the product index. Thus the literal nonresonant-looking factor $\Vert \theta {\Vert }^2=1/4$ cannot give the printed decay in this specialization. Direct integer evaluations at $\lambda =1,2,3$ give respectively $27/27$, $729/729$ and $19683/19683$. The all-scale observation is the classical congruence above, not an inference from these finite checks. It is not claimed as new mathematics, a Lean result, or a counterexample to RH.

For comparison, Nathan Toumi’s arXiv:2504.02784v1, The level of distribution of the sum-of-digits function in arithmetic progressions, retains the condition $\gcd (b,q-1)=1$ for the phase $e(\ell s_q(n)/b)$ with $0<\ell <b$ (Introduction, equation labelled hypob and the Gowers estimate labelled NDGG in the author TeX). The latter estimate has fixed order $k\ge 3$. This condition excludes the base-three digit-parity resonance. Its truncation is explicitly reduction modulo $q^{\rho }$, rather than the inclusive cutoff printed in Jelinek’s (1.1). Toumi’s result is a comparison of source conditions, not a substitute theorem for Fibonacci numeration or the actual FIB input.

Reuse and the remaining same-source obligation

The Zeckendorf digit-parity Möbius orthogonality result is already recorded in FIB §148.2. The prime digit-sum distribution and its modulus restriction are already recorded in the Drmota–Müllner–Spiegelhofer note. Those results are reused; no new proof, replacement scan or claim of priority is supplied here.

The actual input in FIB §384 is the cumulative convolution $e=\mu \ast \beta$, with $H_N={\sum }_{d\le N}{\beta }_d\mathfrak{M}(\lfloor N/d\rfloor )$. A digital phase estimates a different observable. Transport must keep these weights and divisor arguments on the same arithmetic source and must retain the constant mode. Taking an integral phase supplies no digital oscillation; small nonconstant digital correlations do not settle that missing mode. The claimed norm must also belong to the specified group and measure before any Gowers duality is used.

The Rauzy/carry framework is therefore a geometric lead. A repaired, applicable phase statement, its precise group model, and a bound for the required same-source weighted divisor response remain separate obligations. No actual-$H$ square-root growth, signed Robin remainder, or RH proof follows from this source audit.