Computer Science/Discrete Mathematics Seminar I

Over a high-dimensional vector space $F_p^n $over a fixed finite field $F_p$, the inverse theorem for the Gowers norm asserts that a bounded function f on this space that has a large Gowers $U^{k+1}$ 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 $U^6(F_2^n)$ norm correlate with quintic phases, but such phases can be necessarily non-measurable.