Necessity of σ-dependent resilience loss under B_σ product distributions

Ascertain whether the dependence on the parameter σ in the resilience bounds of the Ajtai–Linial-style explicit construction extended to the biased product distribution B_σ is necessary, or whether comparable resilience can be achieved without σ-dependent degradation.

Background

The authors generalize the Ajtai–Linial construction to product distributions B_σ, obtaining explicit circuits with resilience that includes a loss depending on σ (specifically σ-dependent factors in the tradeoff).

Prior work by Filmus, Harsha, Hatami, Hosseini, and Zuckerman (FHHHZ19) shows that the KKL theorem essentially persists under arbitrary product distributions, limiting resilience to roughly n log log n / log n. The paper’s extension to B_σ incurs additional σ-dependent loss, and the authors explicitly note uncertainty about whether this dependence is an inherent limitation or an artifact of the construction.