A Dense Model Theorem for the Boolean Slice

The (low soundness) linearity testing problem for the middle slice of the Boolean cube is as follows. Let ϵ > 0 and f be a function on the middle slice on the Boolean cube, such that when choosing a uniformly random quadruple (x,y, z,x ⊕ y⊕ z) of vectors of 2n bits with exactly n ones, the probability that f(x⊕ y⊕ z)=f(x)⊕ f(y)⊕ f(z) is at least 1/2+ϵ. The linearity testing problem, posed by [6], asks whether there must be an actual linear function that agrees with f on 1/2+ϵ′ fraction of the inputs, where ϵ′=in′(in) > 0. We solve this problem, showing that f must indeed be correlated with a linear function. To do so, we prove a dense model theorem for the middle slice of the Boolean hypercube for Gowers uniformity norms. Specifically, we show that for every k N, the normalized indicator function of the middle slice of the Boolean hypercube 0,1²ⁿ is close in Gowers norm to the normalized indicator function of the union of all slices with weight t=n(mod}\ 2^k-1). Using our techniques we also give a more general 'low degree test' and a biased rank theorem for the slice.