Halfspaces are hard to test with relative error

Several recent works (Chen et al., SODA 2025; Chen et al., ICALP 2025; Chen et al., COLT 2025; Chen et al., manuscript) have studied a model of property testing of Boolean functions under a relative-error criterion. In this model, the distance from a target function \(f : \{0, 1\}^n \rightarrow \{0, 1\}\) that is being tested to a function \(g\) is defined relative to the number of inputs \(x\) for which \(f(x) = 1\); moreover, testing algorithms in this model have access both to a black-box oracle for \(f\) and to independent uniform satisfying assignments of \(f\). The motivation for this model is that it provides a natural framework for testing sparse Boolean functions that have few satisfying assignments, analogous to well-studied models for property testing of sparse graphs.