Order of mixing

Definitions and mathematical statement

Choose a prime $p$ for which \[ f(u,v)=1+uv+u^2v+u^3v+u^4+v^2+u^4v^2 \] is irreducible in $\mathbb F_p[u^{\pm1},v^{\pm1}]$. Let $X$ be the closed subgroup of $\mathbb F_p^{\mathbb Z^2}$ consisting of all configurations such that, for every $(a,b)\in\mathbb Z^2$, \[ x_{a,b}+x_{a+1,b+1}+x_{a+2,b+1}+x_{a+3,b+1} +x_{a+4,b}+x_{a,b+2}+x_{a+4,b+2}=0. \] All sums are in $\mathbb F_p$. Equip $X$ with its normalized Haar probability measure $\mu$, and let $(T^h x)_g=x_{g+h}$ denote the shift action of $\mathbb Z^2$.

The action is mixing of order $k$ if, for every measurable $A_1,\ldots,A_k\subset X$, \[ \mu\!\left(\bigcap_{i=1}^kT^{-h_i}A_i\right) \longrightarrow\prod_{i=1}^k\mu(A_i) \quad\text{as }\min_{i\ne j}\|h_i-h_j\|\longrightarrow\infty. \] Let $M$ be the greatest integer $k$ for which this holds. The source gives $3\le M<7$. Determine the exact value of $M$ for the displayed action, and clarify any dependence on the admissible prime.

The limit allows arbitrary independently varying translation vectors, not just multiples of a single fixed configuration of directions. Thus checking one mixing shape alone does not establish the definition. To identify $M$, one needs both mixing at that order and failure at the next, with the failure witnessed by suitable measurable sets and translations. The finite-field alphabet and Haar law are fixed once the prime has been chosen.

Short English statement

Find the exact order of mixing of the explicit algebraic two-dimensional shift, known to lie between three and six.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4300001 (AMR-042-0001), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Ward - Six problems in Algebraic Dynamics (2006), Problem A. Original source indexed by AMR-042-0001. https://www.imath.kiev.ua/~skolyada/kevin.pdf.