bibkey: badziahinshallit2023kronecker
authors: Dmitry Badziahin; Jeffrey Shallit
year: 2023
title: Badly approximable numbers, Kronecker's theorem, and diversity of Sturmian characteristic sequences
doi: 10.5802/jtnb.1236
url: https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1236/
claim: "The published rotation-gap bound supplies a positive multiplier of at most linear size in the modulus for any fixed golden prefix arc; combined with Gronwall's theorem, every fixed legal Zeckendorf-prefix cylinder has normalized divisor-sum limsup one, without an additive Robin sign or preservation of CA identity."
strata_touched: []
license: citation-only
triage: anchor

Quantitative golden rotation and fixed-prefix Robin saturation

The primary source is Badziahin–Shallit, Journal de théorie des nombres de Bordeaux 35 (2023), no. 1, 1–15, DOI:10.5802/jtnb.1236. The published PDF has 16 PDF pages including its cover. The definitions on printed pp.2–3, Theorem 2.3 and Corollary 2.5 were inspected. This is a source-statement and applicability check, not a complete proof audit or Lean verification. No source text is vendored.

The application below combines that published bound with the existing canonical FIB interface and the classical Grönwall theorem. It introduces no new rotation theorem, maximal-order theorem or mathematical originality claim. A targeted search did not locate a primary publication stating the complete combined prefix conclusion; that bounded search result does not establish novelty.

The published gap supplier

For an irrational real number $\theta$, let $H(\theta ,Q)$ denote the largest circular gap among

$$\{0,\theta ,2\theta ,\dots ,Q\theta \}(\mathrm{mod}1),Q\in {\mathbb{N}}_{>0}.$$

For an integer $B\ge 1$, use $S_B$ here for the irrational members of the source’s set of reals whose continued-fraction partial quotients after the integer part are at most $B$. The source defines

$$f(B)=\underset{\theta \in S_B}{\sup }\underset{Q\ge 1}{\sup }QH(\theta ,Q).$$

Theorem 2.3 gives the exact value of $f(B)$; Corollary 2.5 supplies the bound used here:

$$H(\theta ,Q)\le \frac{f(B)}{Q},f(B)\le (1+\sqrt{4/5})B<2B(\theta \in S_B).\text{\hspace{1em}(P1)}$$

The parameter $B$ must be supplied for the actual angle. In the application that angle is $\{m\phi \}$ and varies with $m$; a fixed bound for the partial quotients of $\phi$ alone is insufficient.

Reuse the exact canonical prefix arc

Set $\phi =(1+\sqrt{5})/2$, $\tau ={\phi }^{-1}$, $\psi =-\tau$ and $F_0=0,F_1=1$. Use canonical digits ${\epsilon }_j(n)$ at weights $F_{j+2}=1,2,3,5,\dots$, with no adjacent occupied digits. The unit digit is position zero. The five window labels occupy later positions and retain their separate unit field and adjacency seams.

The primary digit supplier is Drmota–Müllner–Spiegelhofer, Möbius orthogonality for the Zeckendorf sum-of-digits function, arXiv:1706.09680v1, Lemma 1 on PDF p.4, with the preceding interval definition; the published paper has DOI:10.1090/proc/14015. That lemma characterizes fixed low digits by an interval condition on $(-1{)}^{k}n\phi$ modulo one. Its use is already recorded in the FIB theory, §148.2. The following parameter correspondence also reuses that volume’s Theorems 116.4–116.5 prefix parameterization; it is not inferred from density.

Let $w=(w_0,\dots ,w_{L-1})$ be a fixed nonempty legal prefix. Put

$$V=\sum _{j<L}{w}_j{F}_{j+2},b=w_{L-1},h=L+b,a_w=-\sum _{j<L}{w}_j{\psi }^{j+2},J=(-\tau ,{\tau }^2).$$

Write $C_w$ for the nonnegative integers with that canonical prefix. The existing shift $s$ gives $C_w=V+s^h({\mathbb{N}}_0)$ and

$$\eta (n):=n\phi -s(n)=-\sum _j{\epsilon }_j(n){\psi }^{j+2}\in J,\eta (V+s^h(t))=a_w+{\psi }^h\eta (t).$$

Equivalently, the primary lemma with $k=L+2$ and $u=V$ supplies the circular arc

$$I_w=(a_w+{\psi }^{h}J)(\mathrm{mod}1),|I_w|={\phi }^{-h},n\in C_w⟺\{n\phi \}\in I_w(n\ge 0).\text{\hspace{1em}(P2)}$$

When $b=0$, $V<F_{L+1}$ and the lemma uses its first interval; when $b=1$, $F_{L+1}\le V<F_{L+2}$ and it uses its second interval. The latter interval is shorter by one golden scale, matching the forced zero after the last occupied digit and $h=L+1$.

The two endpoints in (P2), in either order, project to

$$\{(V-F_{h+1})\phi \},\{(V-F_{h+2})\phi \}.$$

Indeed, ${\psi }^r=F_{r+1}-\phi F_r$. Legal occupancy gives $V<F_{h+1}$, so these endpoints are phases of strictly negative integers. Irrationality prevents any $n\ge 0$ from attaining them. Thus the open-arc convention has no finite-source endpoint exception, including at $n=0$. Projection to the circle also handles arcs crossing zero. For $w=(1)$, it gives the existing unit-one interval $I_w=(2-\phi ,2(2-\phi ))$. The empty prefix imposes no restriction and needs no arc construction.

A polynomial-cost multiple in every fixed cylinder

For an integer $r\ge 1$, let $a$ be the nearest integer to $r\phi$. The nonzero quadratic norm is an integer, so the standard golden irrational separation estimate follows from

$$1\le |a^2-ar-r^2|=|r\phi -a||r\psi -a|\le \Vert r\phi \Vert (\sqrt{5}r+\frac{1}{2})<3r\Vert r\phi \Vert .$$

In particular, for $m,q\ge 1$ and ${\theta }_m=\{m\phi \}$,

$$\Vert q{\theta }_m\Vert =\Vert qm\phi \Vert >\frac{1}{3mq}.$$

The usual convergent inequality $\Vert q_j{\theta }_m\Vert <1/(a_{j+1}{q}_j)$ bounds every partial quotient by $3m$; this includes the first noninteger quotient. Use (P1) with $B=3m$, ${\ell }_w=|I_w|$ and $Q=\lceil 6m/{\ell }_w\rceil$. Then

$$H({\theta }_m,Q)<\frac{6m}{Q}\le {\ell }_w.$$

Translating the entire point set by ${\theta }_m$ preserves its circular gaps. Consequently one of ${\theta }_m,2{\theta }_m,\dots ,(Q+1){\theta }_m$ lies in the open arc $I_w$. This supplies a positive multiplier, rather than merely the zero orbit point. With $K_w=6{\phi }^h+2$, there is an integer $k$ such that

$$1\le k\le K_{w}m,H=km\in C_w,m\le H\le K_w{m}^2.\text{\hspace{1em}(P3)}$$

The constant depends on the fixed prefix. This is a direct application of (P1), not a new quantitative Kronecker theorem.

Relative Robin saturation survives every fixed prefix

Write $Z(n)=\sigma (n)/n={\sum }_{d\mid n}{d}^{-1}$ and $R(n)=Z(n)/(e^{\gamma }\log \log n)$ for sufficiently large $n$. The classical Grönwall theorem supplies $m_i\to \infty$ with $R(m_i)\to 1$ and the global upper limsup one. Apply (P3) to each $m_i$. For the same actual integer $H_i=k_i{m}_i\in C_w$, divisor inclusion and the size bound give

$$Z(H_i)\ge Z(m_i),\log m_i\le \log H_i\le 2\log m_i+\log K_w,\frac{\log \log m_i}{\log \log H_i}⟶1.$$

No coprimality assumption on $k_i,m_i$ is needed. Therefore ${\mathrm{liminf}}_{i}R(H_i)\ge 1$, while the global Grönwall upper limsup gives ${\mathrm{limsup}}_{i}R(H_i)\le 1$. The resulting application is

$$\underset{\begin{array}{c}n\to \infty \\ n\in C_w\end{array}}{\mathrm{limsup}}\frac{\sigma (n)}{e^{\gamma }n\log \log n}=1\text{for every fixed legal prefix }w.\text{\hspace{1em}(P4)}$$

In particular, no such cylinder has an eventual bound $R(n)\le 1-{\delta }_w$ for a fixed ${\delta }_w>0$. Every nonempty class determined solely by finitely many fixed low windows and the unit digit contains one of these cylinders and has the same upper limsup. Intersecting a class with CA identity or other prime-factor restrictions changes the population; (P4) supplies no conclusion for that intersection.

These cylinders impose no termination constraint. A certified End immediately after a fully observed prefix fixes a single integer; (P4) supplies no conclusion for additional End or length restrictions.

The additive sign and the original source remain unpaid

For $m>e$ and a multiplier in (P3), the exact additive ledger is

$$\begin{array}{rl}\Delta (km)-\Delta (m)& =e^{\gamma }\log \left(1+\frac{\log k}{\log m}\right)-[Z(km)-Z(m)],\\ \Delta (n)& =e^{\gamma }\log \log n-Z(n).\end{array}$$

The first term is allowed to reach $e^{\gamma }\log 2+o(1)$ by the available size bound; only nonnegativity is known for the bracketed term. Thus (P4) supplies neither additive convergence to zero nor the strict Robin sign. Its guaranteed relative denominator loss is $O_w(1/\log \log m)$, which at $\log m\asymp P$ is $O_w(1/\log P)$, too coarse for the $1/(\sqrt{P}\log P)$ comparison needed at an actual CA host. The construction also supplies no preservation of prime support, ordered exponents or CA identity.

The finer near-zero CA-host comparator is already available in Baker’s source application; it should be reused on its stated branch. The multiplicative Beatty average does not bound an individual extremal integer. The complete Robin comparison, GPP7, still requires the full same-source signed once-prime estimate. Neither (P4) nor a change of FIB coordinates supplies it. The all-integer strict Robin inequality and RH remain unresolved here.