bibkey: fibcharacter2026bohrbarrier authors: trureturing research synthesis year: 2026 title: Bohr phase boxes and weighted characters at a fixed Fibonacci modulus doi: null url: https://arxiv.org/abs/1011.0107v2 claim: “Classical Bohr capacity supplies simultaneous phase alignment. Applied to the full divisor-increment law, it obstructs uniform nonprincipal-character decay at the Robin budget scale; it does not settle actual weighted residue hits.” strata_touched: [] license: citation-only triage: anchor

Bohr phase boxes and the actual residue target

The classical ingredients below are distinguished from the parameter-dependent application in §§220–222 of the FIB theory. The application is a paper derivation, with no Lean verification or originality claim. The fixed-modulus joint-weight interface is also recorded in the complementary-divisor note.

Primary sources and exact scope

Tom Sanders, On the Bogolyubov–Ruzsa lemma, arXiv:1011.0107v2 (2012), original PDF.

Section 5, Lemma 5.1 (PDF p.8) bounds the measure of intersections of Bohr sets. Equation (5.2) and the discussion on p.9 give the one-frequency estimate and explain the usual multiple-frequency lower bound. The paper points to Tao–Vu, Additive Combinatorics, Lemma 4.20, and distinguishes its chord-distance definition from the book’s phase-distance definition. The book is a bibliographic pointer here; the explicit phase-box argument below supplies the constants used in the FIB application.

Jonathan Hermon and Sam Olesker-Taylor, Cutoff for Almost All Random Walks on Abelian Groups, arXiv:2102.02809v2 (2025), original text, §2.6.

Section 2.6, “Lower Bound on Total-Variation Mixing”, proves the lower bound in Theorem 2.5 for every fixed choice of generators. Its proof explicitly fixes the generators instead of averaging over them. A source distribution concentrated on a small set of likely exponent vectors has an image supported on at most that many group elements. Random-generator upper bounds elsewhere in the paper do not establish mixing for the deterministic small-prime residues used here.

The elementary abstract interface is as follows. Let $\rho$ be a probability on a countable source, let $\pi$ map that source to a group of order $Q$, and let

$$E_L=\{w:\rho (w)\ge e^L/Q\}.$$

Then $|E_L|\le Q{e}^{-L}$ and $|\pi (E_L)|\le |E_L|$, so

$$\Vert {\pi }_{\ast }\rho -\mathrm{Unif}{\Vert }_{\mathrm{TV}}\ge \rho (E_L)-e^{-L}.$$

No suitable high-probability source-entropy estimate for the present increment law is supplied here. In particular, this source does not establish that its total variation tends to one.

Finite simultaneous phases

Let $G$ be any finite abelian group of order $Q$, and choose $a_1,\dots ,a_m\in G$. Partition the circle into $L$ equal half-open arcs. The $Q$ characters occupy at most $L^m$ phase boxes at these elements. A largest box contains at least $Q/L^m$ characters. Dividing its characters by one fixed member produces that many distinct characters with

$$\chi (a_j)=e^{i{\theta }_j},|{\theta }_j|\le 2\pi /L(1\le j\le m).$$

This is a near-annihilator Bohr set in the dual group. It includes the principal character; the capacity comparison $Q/L^m>1$ is needed to ensure another one. Neither generation of the group nor independence of the phase coordinates is required.

For the actual FIB application, $V=F_r$ with prime $r\to \infty$, $A=1+V\lceil V/10\rceil$, $y=\log A$, $\ell =\log y$, $s=y\ell$ and $R=y/{\ell }^2$. Every prime divisor of $V$ is at least $2r-1$, so all $p\le 2y$ are units eventually. Here $Q=\phi (V)$, $\log Q=y/2+O(1)$, $m=\pi (2y)=(2+o(1))y/\ell$, and $L=\lfloor y^{1/8}\rfloor$ gives at least $\exp (y/4+o(y))$ aligned characters.

The actual increment law is

$$b_s(1)=1,b_s(p^a)=Z(p^a{)}^s-Z(p^{a-1}{)}^s,{\mu }_s(d)=\frac{b_s(d)}{dU(s)},Z(n)=\frac{\sigma (n)}{n}.$$

Its unit-conditioned pushforward is ${\nu }_s$; for Dirichlet characters extended by zero off the units, ${\hat{\mu }}_s(\chi )=c_V{\hat{\nu }}_s(\chi )$, where $c_V={\mu }_s((d,V)=1)$. Section 222 combines the phase count with the genuine exponent second moment and the large-prime tail to derive

$$|{\hat{\nu }}_s(\chi )|,|{\hat{\mu }}_s(\chi )|\ge \exp (-C{y}^{3/4}\ell )=\exp (-o(R))$$

for exponentially many nonprincipal characters. The classical Bohr count alone does not give this weighted estimate; the local increment and tail controls are necessary.

The existing Kronecker note describes torus orbit closures, not this growing finite character count. The existing PrimeOnlyNoGap source treats a fixed summable jump law and integer Fourier modes. A return there may be trivial after passage to a finite quotient; it does not by itself give the nonprincipal characters or their number required here.

What the actual kernel asks of the characters

Take the same actual interval $I=[g_{-},g_{+}]\cap \mathbb{Z}$, with $A=1+g_{-}V$, $X=1+g_{+}V$ and $g_{+}-g_{-}<V$. Let $V<D<X$, $H=X/D<V$, and $Q=\phi (V)$. For each positive integer $h\le H$ set

$$J_h=(D,X]\cap [A/h,X/h]\cap \mathbb{N}.$$

For nonnegative weights $a(d)$, define the actual hit sum

$$S_a=\sum _{\begin{array}{c}d>D\\ \exists g\in I:d\mid 1+Vg\end{array}}a(d).$$

Each divisor hits at most once: a hit makes $d$ a unit modulo $V$, and $d>V>|I|-1$. For the actual complementary factor $h$, character orthogonality therefore gives

$$S_a=\frac{1}{Q}\sum _{\chi \mathrm{mod}V}\sum _{\begin{array}{c}1\le h\le H\\ (h,V)=1\end{array}}\chi (h)A_{\chi }(J_h),A_{\chi }(J)=\sum _{\begin{array}{c}d\in J\\ (d,V)=1\end{array}}a(d)\chi (d).$$

The retained kernel uses $a(d)=(d/X){\mu }_s(d)$. Additional restrictions on the same product $dh$ must stay inside the joint sum; a condition depending on $dh$ cannot automatically be absorbed into $a(d)$ alone.

Writing $M_0=Q^{-1}{\sum }_h{A}_1(J_h)$, the exact target $S_a<\eta$ is equivalent to

$$\mathrm{Re}\sum _{\chi \ne 1}\sum _h\chi (h)A_{\chi }(J_h)<Q(\eta -M_0).$$

It asks for a signed aggregate with moving intervals. A uniform bound on every character is stronger. Large moduli of the full Euler coefficients do not imply a large mass at residue one, an inverse residue, or an actual multiplier. Conversely, truncation to $J_h$, a cap on $dh$, and the $d/X$ kernel destroy the simple full Euler-product identification. Neither upper nor lower modulus estimates for that product transfer without a separate argument.

A fixed-modulus short-factor fourth moment

For a fixed unit subset $\mathcal{H}\subseteq [1,H]\cap \mathbb{N}$ with $H^2<V$, put $B_{\chi }={\sum }_{h\in \mathcal{H}}\chi (h)$. Orthogonality gives

$$\frac{1}{Q}\sum _{\chi }|B_{\chi }{|}^4=\#\{h_1{h}_2\equiv h_3{h}_4(\mathrm{mod}V):h_i\in \mathcal{H}\}.$$

Both products lie in $[1,H^2]$, so each congruence is an integer equality. Write $h_1=gu$, $h_3=gv$ with $(u,v)=1$; equality then forces $h_2=kv$, $h_4=ku$. Consequently

$$\frac{1}{Q}\sum _{\chi }|B_{\chi }{|}^4\le \sum _{\begin{array}{c}u,v\le H\\ (u,v)=1\end{array}}{\lfloor \frac{H}{\max (u,v)}\rfloor }^2\le 2H^2(1+\log H).$$

This elementary multiplicative-energy estimate needs no prime-modulus assumption. For a fixed interval $J$ and $A_{\chi }=A_{\chi }(J)$, Hölder yields

$$\left|\frac{1}{Q}\sum _{\chi \ne 1}{A}_{\chi }{B}_{\chi }\right|\le {\left(\frac{1}{Q}\sum _{\chi \ne 1}|A_{\chi }{|}^{4/3}\right)}^{3/4}{\left(2H^2(1+\log H)\right)}^{1/4}.$$

The core spectral moment has not been bounded at the required scale. The original intervals $J_h$ also move with $h$; using fixed rectangles requires explicit coverage and boundary errors. Enlarging a nonnegative hit set gives an upper bound, not an equality. These standard identities identify a possible aggregate estimate without supplying it or solving Robin for the full FIB family.