The paper studies agreement theorems for high dimensional expanders in the low acceptance (or 1%) regime. Agreement tests are used to determine if an ensemble of local functions {fs : s → Σ | s ∈ X} on a simplicial complex X can be "explained" by a global function G : [n] → Σ.
The main results are:
If the complex X has no connected covers, then the classical 1% agreement theorem holds, provided X satisfies an additional property called swap cosystolic expansion.
If X has a connected cover, then the classical 1% agreement theorem fails.
If X has a connected cover (and satisfies swap-cosystolic-expansion), a weaker agreement theorem is shown to hold. This theorem takes the cover structure into account, showing that the ensemble {fs} can be "explained" by a global function G defined on a cover Y of X.
The paper also shows that many known constructions of sparse high dimensional expanders, including spherical buildings and LSV complexes, satisfy the required properties to obtain these agreement theorems. This improves upon previous results, providing the sparsest known families of complexes that support 1% agreement theorems.
The technical approach involves constructing compatible lists of local functions on the faces of the complex, using the expansion properties to resolve local inconsistencies, and then lifting these to a global function on a cover of the complex.
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