math.NT — Number Theory

The Welch function is not third-order sum-free

Contributed by Deep Bhattacharjee

Let $n=2m+1$. Hou and Zhao conjectured that for $n\ge7$ the Welch function $X^{2^m+3}$ on $\F_{2^n}$ sums to zero over some three-dimensional affine subspace, that is, it is not third-order sum-free. We prove this. The proof counts exactly the elements $x$ for which some subspace spanned by $1$, $x$ and a third element gives a zero sum: there are $(2^n+1-3K_n)/4$ of them, where $K_n$ is a binary Kloosterman sum.