Unbounded Violations of the Square-Root Degree Bound

We disprove the Gopalan–Servedio square-root conjecture, even up to an arbitrary constant factor. For every real C > 0, there is a nonconstant Boolean function $f:\{-1,1{\}}^n\to \{-1,1\}$ on a finite sign cube such that $\sum _{i=1}^n\hat{f}(\{i\})>C\sqrt{\mathrm{deg}(f)}.$ Here $\hat{f}(\{i\})$ is the linear Fourier coefficient associated with the ith input, and $\mathrm{deg}(f)$ is the degree of the real multilinear polynomial representing f.