Sarnak's Möbius Disjointness Conjecture

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.

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

Current frontier and scope.

The 2026 infinite-dimensional torus paper addresses its specified torus class [S1], not every zero-entropy system. Frantzikinakis--Host prove a logarithmic version for a broad class including uniquely ergodic zero-entropy systems; its weighting and hypotheses cannot be discarded [E1]. The familiar Chowla-to-Sarnak implication motivates the attempt, but here its ordinary mixed-power arithmetic assumption is written explicitly, rather than confused with an already established logarithmic theorem or the exact Liouville statement of Section 77.

Attack route.

Topological zero entropy gives subexponentially many orbit blocks, suggesting a covering argument. A second-moment estimate alone can be overwhelmed by that many blocks. The route pursued combines the covering with arbitrarily high fixed moments, choosing the moment order after the block length but before the averaging limit. This produces an explicit entropy--moment tradeoff and identifies a sufficient arithmetic input. A sparse-set alignment proposal is also checked against the density normalization.

Attempt: reducing every orbit to finitely many blocks.

Fix a continuous complex-valued $f$ and scale so that $\|f\|_\infty\leq1$; the zero function is immediate. For a block length $L$ and $\varepsilon>0$, choose a finite set $\mathcal V_{L,\varepsilon}\subset{\mathbb{C}}^L$ of actual observable blocks such that every \[ (f(Ty),\ldots,f(T^Ly)),\qquad y\in X, \] is within $\varepsilon$ in supremum norm of a member. Put $M_{L,\varepsilon}=|\mathcal V_{L,\varepsilon}|$. Uniform continuity of $f$ and zero topological entropy give \begin{equation}\label{wsref-0-65-71-38-34-3a-65-6e-74-72-6f-70-79} \log M_{L,\varepsilon}=o(L)\quad(L\to\infty) \end{equation} for every fixed $\varepsilon$. This is a covering-number statement for all points of the compact space, not an almost-everywhere orbit theorem.

For every $x$, sliding the length-$L$ window gives \begin{equation}\label{wsref-0-65-71-38-34-3a-63-6f-76-65-72} \left|\frac1N\sum_{n=1}^N\mu(n)f(T^nx)\right| \leq\varepsilon+ \frac1N\sum_{m=0}^{N-1}\max_{v\in\mathcal V_{L,\varepsilon}} \left|\frac1L\sum_{j=1}^L\mu(m+j)v_j\right| +O(L/N). \end{equation} To check the boundary error, average the original products over all windows. Each interior index is counted exactly $L$ times, and the two ends together contain at most $2L$ indices with smaller or extra weight. Their summands have absolute value at most one. Replacing each orbit block by its net representative changes its normalized sum by at most $\varepsilon$. This proves (eq84:cover) with a bound independent of the chosen $x$.

Attempt: an explicit arithmetic hypothesis and moment bound.

Assume the following ordinary mixed-power Möbius Chowla statement: for every finite collection of distinct nonnegative shifts $h_i$ and exponents $e_i\in\{1,2\}$, with at least one $e_i=1$, \begin{equation}\label{wsref-0-65-71-38-34-3a-63-68-6f-77-6c-61} \frac1N\sum_{m=1}^N\prod_i\mu(m+h_i)^{e_i}\longrightarrow0. \end{equation} This is an additional unresolved hypothesis, not an established consequence of the logarithmic results. Its role in the classical implication is consistent with the averaging distinctions in [E1] and [E2].

For fixed $L$, integer $r\geq1$, and $|v_j|\leq1$, expand the $2r$th power of $|\sum_{j=1}^L\mu(m+j)v_j|$. Any term in which some index appears an odd number of times has mean tending to zero by (eq84:chowla), since odd positive powers of $\mu$ equal $\mu$ and even positive powers equal $\mu^2$. A surviving term must have all multiplicities even, and its coefficient and summand each have absolute value at most one. The number of ordered $2r$-tuples with all multiplicities even is at most \[ (2r-1)!!\,L^r. \] Indeed, pair the $2r$ positions in $(2r-1)!!$ ways and give each pair an index in $\{1,\ldots,L\}$. Every even-multiplicity tuple is generated at least once, so overcounting gives an upper bound. It follows that \begin{equation}\label{wsref-0-65-71-38-34-3a-6d-6f-6d-65-6e-74} \limsup_{N\to\infty}\frac1N\sum_{m=0}^{N-1} \left|\frac1L\sum_{j=1}^L\mu(m+j)v_j\right|^{2r} \leq\frac{(2r-1)!!}{L^r}. \end{equation} There are only finitely many terms for fixed $L,r$, so no uniformity in an increasing number of shifts has been assumed in taking this limit. The index $m=0$ changes the average by $O(1/N)$ only.

Attempt: choosing the moment order without exchanging limits.

For a finite net of size $M$, the maximum of the $2r$th powers is at most the sum of them. Hölder's inequality in $m$, followed by (eq84:moment), turns (eq84:cover) into \begin{equation}\label{wsref-0-65-71-38-34-3a-74-72-61-64-65-6f-66-66} \limsup_{N\to\infty}\left|\frac1N\sum_{n=1}^N \mu(n)f(T^nx)\right| \leq\varepsilon+ M_{L,\varepsilon}^{1/(2r)} \frac{((2r-1)!!)^{1/(2r)}}{\sqrt L} \leq\varepsilon+ \exp\left(\frac{\log M_{L,\varepsilon}}{2r}\right) \sqrt{\frac{2r}{L}}. \end{equation} For fixed $\varepsilon$, take $r=\lceil\sqrt{L\max(1,\log M_{L,\varepsilon})}\rceil$. By (eq84:entropy), $r/L\to0$ and $\log M_{L,\varepsilon}/r\to0$. The final term in (eq84:tradeoff) tends to zero. Thus the limsup is at most $\varepsilon$, and then $\varepsilon\downarrow0$ proves the target under (eq84:chowla).

The order of operations is essential. For each selected finite $L$ the integer $r$ and finite net are fixed before $N\to\infty$. Only after that limit do we increase $L$. Consequently the proof does not assume a Chowla error term uniform in $r$ or $L$. The zero-entropy cover uses all points, so the resulting assertion holds for every $x$, not merely almost every point of an invariant measure. Invertibility of $T$ was never used.

Stress test: what weaker data fail to provide.

With $r=1$, the same argument gives an error at best $\sqrt{M_{L,\varepsilon}/L}$, which need not vanish. For instance $M_L=\exp(\sqrt L)$ is consistent with the subexponential bound (eq84:entropy). Thus zero entropy and a second moment do not by this argument suffice; the freedom to use increasing finite moment orders is doing real work. A potentially weaker route than full (eq84:chowla) would prove (eq84:moment) only for the orbit nets and chosen orders required by (eq84:tradeoff), with a similarly controlled constant. No such uniform arithmetic-net estimate is obtained.

The source about arbitrary prescriptions along a sparse set does not give a counterexample. If a bounded observable is chosen to correlate perfectly on $S\subset{\mathbb{N}}$ of density zero, that part contributes at most $|S\cap[1,N]|/N\to0$. Positive-density correlation would require additional structure not supplied by sparse freedom. Likewise logarithmic cancellation in (eq84:chowla) would only support a suitably logarithmic version of the argument; it is not ordinary cancellation. The exact Liouville linear-form assertion in Section 77 is not silently identified with the mixed-power Möbius hypothesis without a separate transfer theorem.

The companion script verifies the combinatorial moment bound by enumerating independent sign sequences for small $L,r$. This checks the pairing count; it does not test the required arithmetic cancellation of shifted Möbius values at all scales.

Outcome.

Outcome: Conditional reduction.

An explicit covering/high-moment proof derives every-point ordinary Möbius disjointness from the stated mixed-power correlation hypothesis. It records the entropy--moment tradeoff and the order of limits, and explains why second moments or sparse-set prescriptions alone do not close the problem. This reconstructs a classical conditional implication rather than claiming a new resolution or novelty for that implication.

The missing step is (eq84:chowla), or the weaker net-specific high-moment input just isolated, for the actual arithmetic sequence. No new unconditional zero-entropy class or counterexample to the original Sarnak assertion is produced.

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. Catalogue 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. Catalogue research/status reference; not a proof endorsement.
• E1 Nikos Frantzikinakis and Bernard Host, The logarithmic Sarnak conjecture for ergodic weights, Annals of Mathematics 187 $2018$, 869--931, main theorem and averaging conventions. https://annals.math.princeton.edu/2018/187-3/p06. Inspected 27 September 2026; the logarithmic weighting and system hypotheses are not discarded.
• E2 Terence Tao, Equivalence of the logarithmically averaged Chowla and Sarnak conjectures, arXiv:1605.04628, introduction and comparison with the ordinary Chowla-to-Sarnak implication. https://arxiv.org/abs/1605.04628. The ordinary mixed-power implication used here is proved in full in the notebook; this reference is not treated as a proof of its unresolved ordinary arithmetic hypothesis.