gpt 6 astra ultra attempt on Math-89

Sarnak's Möbius Disjointness Conjecture

Definitions and mathematical statement

The Möbius function is $\mu(n)=0$ when a prime square divides $n$, and $\mu(n)=(-1)^r$ when $n$ is a product of $r$ distinct primes. A compact metric system consists of a continuous map $T:X\to X$; zero topological entropy means its orbit complexity has zero exponential growth rate. The assertion is \[ \frac1N\sum_{n=1}^{N}\mu(n)f(T^nx)\longrightarrow0 \qquad(f\in C(X),\ x\in X). \] Every point is quantified, not just almost every point under a chosen invariant probability measure. The test functions may be complex-valued.

Formulation and concept references: [S1].

Short English statement

Is the Möbius function asymptotically uncorrelated with every continuous observable of every zero-entropy dynamical system?

Research attempt

Scope and checked context.

This unresolved attempt by GPT-6 Astra Ultra was prepared on 27 September 2026. The quantifier over every point is retained throughout. The concrete subcase proved here is translation on a finite or countably infinite torus, using the classical Möbius exponential-sum theorem. A second argument proves a random-sign analogue for a fixed zero-entropy system. Comparing the two arguments identifies the arithmetic input that entropy alone does not give. These established methods and elementary consequences are not claimed as new results about the conjecture.

The current source [S1] studies a specified family of smooth skew systems on an infinite-dimensional torus; its statement is not a theorem about all continuous zero-entropy systems. The external arithmetic input used below is Davenport's estimate, recorded in the primary research source [E1]: with $e(t)=e^{2\pi it}$, for every $A>0$, \[ \sup_{\theta\in\mathbb R} \left|\sum_{n\leq N}\mu(n)e(n\theta)\right| \ll_A N(\log N)^{-A}. \tag{84.1} \] It is a theorem about actual Möbius weights, not an independence assumption.

An approximation lemma that preserves every point.

For a bounded sequence $a(n)$ and $|w(n)|\leq1$, put $C_N(w,a)=N^{-1}\sum_{n\leq N}w(n)a(n)$. If $\sup_n|a(n)-b(n)|\leq\varepsilon$, then \[ |C_N(w,a)-C_N(w,b)|\leq\varepsilon. \tag{84.2} \] There is no exceptional set and no invariant measure in this estimate. Thus uniform approximation of observables by a class for which disjointness is known proves disjointness for its uniform closure. Approximation in $L^2$ of a chosen invariant measure does not yield $84.2$ along an arbitrary specified orbit; an exceptional point can be outside the set where the approximation is useful.

A full rotation subcase, including infinitely many coordinates.

Let $X=\mathbb T^{\mathbb N}$ with the metric \[ d(x,y)=\sum_{j\geq1}2^{-j}\|x_j-y_j\|, \qquad T(x)=x+\alpha \] for an arbitrary fixed $\alpha\in X$. The map is an isometry. Therefore the orbit metric $d_N(x,y)=\max_{0\leq n<N}d(T^nx,T^ny)$ equals $d(x,y)$. For each fixed separation scale its maximal separated-set size is bounded independently of $N$, by compactness. This proves zero topological entropy directly, including rational or dependent coordinates of $\alpha$.

For a continuous complex-valued $f$ and $\varepsilon>0$, uniform continuity first allows the tail coordinates to be fixed: choose $J$ such that \[ |f(x)-f(x_1,\ldots,x_J,0,0,\ldots)|<\varepsilon \quad\text{for all }x. \] Trigonometric polynomial approximation on $\mathbb T^J$ then gives $P(x)=\sum_{m\in F}c_m e(m\cdot x)$, with a finite set $F\subset\mathbb Z^J$, and $\|f-P\|_\infty<2\varepsilon$. Along the rotation, \[ P(T^nx)=\sum_{m\in F}c_m e(m\cdot x)e(nm\cdot\alpha). \] Equations $84.1$--$84.2$ imply \[ \sup_{x\in X}\left|\frac1N\sum_{n\leq N}\mu(n)f(T^nx)\right| \leq2\varepsilon+O_A\left( \frac{\sum_{m\in F}|c_m|}{(\log N)^A}\right). \tag{84.3} \] The polynomial is fixed before $N\to\infty$. Letting $N$ tend to infinity and then $\varepsilon$ to zero proves the conclusion uniformly over all $x$, a stronger quantifier than needed in this subcase. Finite-dimensional rotations are included. The essential additional structure is finite Fourier approximation of an orbit, not merely its zero entropy.

What entropy does supply: a subexponential orbit cover.

Now let $(X,T)$ be any compact metric zero-entropy system and normalize $\|f\|_\infty\leq1$. For a fixed $\varepsilon>0$, consider its observable orbit vectors \[ \mathcal V_N=\{(f(Tx),\ldots,f(T^Nx)):x\in X\} \subset\mathbb C^N. \] Uniform continuity of $f$ and a spanning set in the orbit metric supply a sup-norm $\varepsilon$-net $\mathcal A_N\subset\mathcal V_N$ with \[ \log|\mathcal A_N|=o_\varepsilon(N).\tag{84.4} \] For precision, choose a metric scale whose closeness implies observable closeness, cover orbit segments of length $N+1$, and choose an actual orbit from each occupied cover element. Changing the metric scale by a fixed factor preserves the zero-entropy conclusion.

For any fixed deterministic bounded weights $w(n)$, the net gives \[ \sup_x|C_N(w,f(T^\cdot x))| \leq\varepsilon+ \max_{a\in\mathcal A_N}\left|\frac1N\sum_{n\leq N}w(n)a_n\right|. \tag{84.5} \] The covering cardinality alone provides no estimate for the last maximum. In particular $84.1$ applies to linear exponential phases, not to an arbitrary vector representing an orbit segment. Calling those vectors "deterministic" does not put them into its hypotheses.

A probabilistic comparison with a complete proof.

Replace Möbius by independent random signs $\epsilon_n$, each taking $1$ and $-1$ with equal probability. For a fixed vector $a$ with $|a_n|\leq1$, Hoeffding's inequality applied to real and imaginary parts gives \[ \mathbb P\left(\left|N^{-1}\sum_{n\leq N}\epsilon_na_n\right|>\delta\right) \leq4\exp(-N\delta^2/4). \tag{84.6} \] Indeed magnitude greater than $\delta$ forces one of those parts to have magnitude greater than $\delta/\sqrt2$. Each part is a sum of independent centered variables in intervals of length at most two. The real tail bound $2\exp(-N u^2/2)$ gives the stated constant. One may derive that real bound directly: independence and $\cosh v\leq\exp(v^2/2)$ give $\mathbb E\exp(t\sum_n\epsilon_n\operatorname{Re}a_n) \leq\exp(Nt^2/2)$. Markov's inequality with $t=u$, and then the same argument for the negative tail, proves it.

Take a union bound over $\mathcal A_N$. By $84.4$, for fixed $\varepsilon,\delta>0$ and large $N$ the resulting probability is at most $4\exp(-N\delta^2/8)$, which is summable. Borel--Cantelli and $84.5$ show that almost surely the limsup of the uniform orbit correlation is at most $\varepsilon+\delta$. Taking a countable sequence of positive $\varepsilon,\delta$ tending to zero proves \[ \sup_{x\in X}\left|N^{-1}\sum_{n\leq N}\epsilon_n f(T^nx)\right| \longrightarrow0\quad\text{almost surely}.\tag{84.7} \] For a fixed system, a countable uniformly dense family in $C(X)$ and $84.2$ even give one probability-one event valid for all continuous observables and all points. This statement fixes the system before drawing the signs. It makes no simultaneous claim over all possible systems.

This successful random-sign proof isolates its unavailable Möbius step: there is no probability space in which the fixed values $\mu(n)$ are the independent centered variables of $84.6$. Möbius also vanishes at nonsquarefree integers and satisfies multiplicative identities. Entropy cannot itself justify applying a probabilistic tail estimate to these specific arithmetic weights.

Sparse coding and exceptional points: two stress tests.

If two bounded sequences $a,b$ differ only on $S\subset\mathbb N$ and $|a_n|,|b_n|\leq1$, then \[ |C_N(\mu,a)-C_N(\mu,b)| \leq2\,\frac{|S\cap[1,N]|}{N}. \tag{84.8} \] When $S$ has asymptotic density zero, arbitrary prescriptions on $S$ cannot create a nonzero limiting correlation from a disjoint sequence. Thus flexibility along a sparse set, as studied in [S2], requires careful interpretation before it can be used against this conjecture. Equation $84.8$ concerns the density-zero case explicitly and does not assert that every possible definition of sparsity is identical.

Conversely, an almost-everywhere theorem under one invariant measure does not automatically cover every point. A compact system can have disjoint invariant components, and a measure supported on one component assigns zero mass to the other. For example the identity map on two points has a point-mass invariant measure that misses the other point entirely. This example does not violate Möbius disjointness; it falsifies the proposed logical upgrade from one measure's almost-everywhere statement to an assertion about all points. The rotation proof and the random-sign proof avoid that upgrade by taking the supremum over $x$ before passing to the limit.

Verification and remaining gap.

The local exact checks verify finite periodic Fourier decompositions, the correlation perturbation bound $84.8$, and the complex tail-bound constants algebraically. These are finite bookkeeping checks, not an independence test for Möbius. The torus conclusion depends explicitly on the established theorem $84.1$; the random-sign comparison is proved from independence, a union bound, and summability.

The first missing step for a general zero-entropy system is control of the arithmetic maximum in $84.5$ for its subexponential orbit covers. The Fourier argument handles a structured subclass; the probabilistic argument explains why a random model succeeds but supplies no theorem for Möbius on arbitrary covers. No exceptional points have been discarded and no random model has been substituted in the final claim. Sarnak's conjecture remains UNRESOLVED.

Sources

• S1 Q. Liu, J. Ma, and H. Wang, The Mobius Disjointness Conjecture on infinite-dimensional torus, Introduction $2026$. https://arxiv.org/abs/2603.11087. Formulation source.
• S2 Minimal zero entropy subshifts can be unrestricted along any sparse set. https://www.cambridge.org/core/journals/ergodic-theory-and-dynamical-systems/article/minimal-zero-entropy-subshifts-can-be-unrestricted-along-any-sparse-set/1357B8C6E5361EC6439CEA52634BD7A0. Further reference; not a proof endorsement.

• E1 Ben Green and Terence Tao, Quadratic uniformity of the Möbius function, arXiv:math/0606087, Example 3 and Section 5. https://arxiv.org/pdf/math/0606087. Primary research source recording and proving the classical Davenport estimate used for torus rotations. Its stronger nilsequence results are not needed in this attempt. Consulted 27 September 2026.
• E2 Qingyang Liu, Jing Ma, and Hongbo Wang, The Möbius Disjointness Conjecture on infinite-dimensional torus, arXiv:2603.11087v1 (11 March 2026). https://arxiv.org/abs/2603.11087v1. Version-specific identification of [S1]; the inspected abstract specifies a family of smooth skew systems, not all zero-entropy systems. The paper's theorem is not used in the proofs here. Consulted 27 September 2026.