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A skew lattice construction for the even-radius sufficiency direction

Abstract

A uniform sufficient family of signed two-generator Cayley diameter bounds.

Theorem 1.1 (Distinct spanning pairs for k = 2(t² − j)).

Lean statement: D5/S3/FiniteGroups/BajnokBeattyCayleySufficiency.result

Proof. Machine-checked in Lean as D5/S3/FiniteGroups/BajnokBeattyCayleySufficiency.result (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Bela Bajnok and W. Kyle Beatty (2025). On the diameter of undirected Cayley graphs of finite abelian groups. DOI: 10.2140/cnt.2025.14.1. URL: https://arxiv.org/abs/2406.04045v1.

Commentary.

For every natural t ≥ 1 and 0 ≤ j < t, assume t is even or j is even. Put k = 2(t² − j). The group Z/2 × Z/(2k) contains distinct elements a,b such that every g equals x•a + y•b for integers x,y with |x| + |y| ≤ 2t. Thus the two-element generating set has undirected Cayley diameter at most 2t. The statement includes t=1,j=0 and extends past the top strip 0 ≤ 2j < t.

The covering mechanism is simultaneous remainder descent for the skew vectors (A,−b) and (−a,A), where A>0 and 0≤a,b<A. A positive multiple of their sum moves a coset representative into the nonnegative quadrant. A representative of least coordinate sum must lie in [0,A)², because subtracting a basis vector otherwise preserves nonnegativity and strictly reduces that sum. Applied after integer rotation to A=4t,a=4j,b=4, the square bound becomes a signed coefficient diamond of radius 2t. The endpoint uses integer division directly.

The lattice relations are twice (t−j,−t−j) and twice (t−1,t+1). Explicit surjective additive maps into Z/2 × Z/(4(t²−j)) annihilate both relations. For even t the coordinate change has determinant −1. For odd t and even j, d=t²−j is odd and a Bézout identity for 4 and d lifts every group element. The two coordinate images are distinct because their first coordinates sum to one in Z/2. Translation by the group identifies this signed coefficient ball with the same radius bound from every vertex.

This is partial sufficiency for Bajnok–Beatty Conjecture 4.1 in the checked arXiv v1 scope. General necessity, including exclusion of arbitrary alternative primitive lattices, remains open. No conclusion about the full conjecture or literature priority follows from the construction.

References