Non-measurability of the inverse theorem for the Gowers norms

Over a high-dimensional vector space Fnpover a fixed finite field Fp, the inverse theorem for the Gowers norm asserts that a bounded function f on this space that has a large Gowers Uk+1 norm, must necessarily correlate with a phase polynomial e(P) of degree k this result has a number of applications in additive combinatorics and property testing.  In the high-characteristic case p>=k−1 it is known that this phase polynomial e(P) is "measurable" in the sense that it can be approximated to high accuracy by a function of a bounded number of random shifts of f.  In joint work with Asgar Jamneshan and Or Shalom, we show that this measurability fails when p=2 and k=5, thus functions of large U6(Fn2) norm correlate with quintic phases, but such phases can be necessarily non-measurable.