OAI.SquareRootDegree.main

The theorem, currently admitted without proof, states two things about real Fourier analysis on Boolean cubes, where a Boolean input is read as a sign (false is +1, true is -1), averages are uniform, and degree is the ordinary real Fourier degree, meaning the largest size of a subset s with nonzero Fourier coefficient, the coefficient being the average of f(x) times the product of the signs of the coordinates in s. First, SignedViolations holds: for every real C>0 there is a positive integer n and a function f from {false,true}^n to the reals that takes only the values -1 and 1, is nonconstant, and satisfies C·√(deg f) < Σᵢ f̂({i}), the plain signed sum of its degree-one Fourier coefficients over single coordinates. Second, the set of ratios (Σᵢ |f̂({i})|)/√(deg f), taken over all positive n and all Boolean-valued f on n coordinates with positive Fourier degree, is not bounded above in the reals, so the supremum of these ratios is infinite.