An Efficient Regularity Lemma for Semi-Algebraic Hypergraphs

We use the polynomial method of Guth and Katz to establish stronger and {\it more efficient} regularity and density theorems for such $k$-uniform hypergraphs $H=(P,E)$, where $P$ is a finite point set in ${\mathbb R}^d$, and the edge set $E$ is determined by a semi-algebraic relation of bounded description complexity. In particular, for any $0<\epsilon\leq 1$ we show that one can construct in $O\left(n\log (1/\epsilon)\right)$ time, an equitable partition $P=U_1\uplus \ldots\uplus U_K$ into $K=O(1/\epsilon^{d+1+\delta})$ subsets, for any $0<\delta$, so that all but $\epsilon$-fraction of the $k$-tuples $U_{i_1},\ldots,U_{i_k}$ are {\it homogeneous}: we have that either $U_{i_1}\times\ldots\times U_{i_k}\subseteq E$ or $(U_{i_1}\times\ldots\times U_{i_k})\cap E=\emptyset$. If the points of $P$ can be perturbed in a general position, the bound improves to $O(1/\epsilon^{d+1})$, and the partition is attained via a {\it single partitioning polynomial} (albeit, at expense of a possible increase in worst-case running time). In contrast to the previous such regularity lemmas which were established by Fox, Gromov, Lafforgue, Naor, and Pach and, subsequently, Fox, Pach and Suk, our partition of $P$ does not depend on the edge set $E$ provided its semi-algebraic description complexity does not exceed a certain constant. As a by-product, we show that in any $k$-partite $k$-uniform hypergraph $(P_1\uplus\ldots\uplus P_k,E)$ of bounded semi-algebraic description complexity in ${\mathbb R}^d$ and with $|E|\geq \epsilon \prod_{i=1}^k|P_i|$ edges, one can find, in expected time $O\left(\sum_{i=1}^k|P_i|\log (1/\epsilon)+(1/\epsilon)\log(1/\epsilon)\right)$, subsets $Q_i\subseteq P_i$ of cardinality $|Q_i|\geq |P_i|/\epsilon^{d+1+\delta}$, so that $Q_1\times\ldots\times Q_k\subseteq E$.