Let $p$ be a fixed prime. A $k$-cycle in $\mathbb{F}_p^n$ is an ordered $k$-tuple of points that sum to zero; we also call a $3$-cycle a triangle. Let $N=p^n$, (the size of $\mathbb{F}_p^n$). Green proved an arithmetic removal lemma which says that for every $k$, $\epsilon>0$ and prime $p$, there is a $\delta>0$ such that if we have a collection of $k$ sets in $\mathbb{F}_p^n$, and the number of $k$-cycles in their cross product is at most a $\delta$ fraction of all possible $k$-cycles in $\mathbb{F}_p^n$, then we can delete $\epsilon N$ elements from the sets and remove all $k$-cycles. This is closely related to the graph removal lemma, which essentially says that if a graph $G$ has few copies of a fixed subgraph $H$, then we can remove a small number of edges from $G$ and get rid of all copies of $H$. Green posed the problem of improving the quantitative bounds on the arithmetic triangle removal lemma, and, in particular, asked whether a polynomial bound holds. Despite considerable attention, prior to our work, the best known bound for any $k$, due to Fox, showed that $1/\delta$ can be taken to be an exponential tower of twos of height logarithmic in $1/\epsilon$ (for a fixed $k$).
In this talk, we will discuss recent work on Green’s problem. For triangles, we prove an essentially tight bound for Green’s arithmetic triangle removal lemma in $\mathbb{F}_p^n$, using the recent breakthroughs with the polynomial method. For $k$-cycles, we also prove a polynomial bound. We also prove a lower bound on the exponent by proving a lower bound on the $k$-multicolored sum-free problem. However, the question of the optimal exponent is still open.
The triangle case is joint work with Jacob Fox, the $k$-cycle case with Jacob Fox and Lisa Sauermann, and the lower bound for general $k$ with Lisa Sauermann.