Degree-1 Fourier Weight Conjecture for Linear Threshold Functions

Prove that every Boolean linear threshold function on a finite Hamming cube satisfies the lower bound [?]

Background

The paper studies the degree-0 and degree-1 Fourier weight of Boolean linear threshold functions (LTFs), denoted by W{\leq 1}[f]. The authors establish the explicit lower bound W{\leq 1}[f] > 0.53317, improving the classical 1/2 bound but not reaching 2/\pi.

The unresolved target is O'Donnell's Degree-1 Fourier Weight Conjecture, which predicts the sharp universal lower bound 2/\pi. The paper explicitly presents its result as progress toward this conjecture, so the conjecture remains unresolved in the paper.

References

O'Donnell's book Conjecture 5.3 put forward the Degree-$1$ Fourier Weight Conjecture, also referred to as the Benjamini--Kalai--Schramm conjecture in some contexts: every LTF $f$ should satisfy

\mathbf{W}{\leq1}[f]\geq \frac{2}{\pi}.

— A Two-regime Khintchine Inequality and an Improved Bound on the Degree-1 Fourier Weight for Linear Threshold Functions  (2608.27908 - Fang et al., 28 Aug 2026) in Section 1, Introduction; Conjecture 5.3 of O'Donnell cited in Section 1