A note on lower bounds for arithmetic regularity partitions

Abstract: This paper establishes lower bounds for two kinds of arithmetic regularity partitions, building on constructions of Green [arXiv:math/0310476v2] and Hosseini, Lovett, Moshkovitz, and Shapira [arXiv:1405.4409]. The first kind occurs in the so-called strong arithmetic regularity lemma due to Bhattcharrya, Fischer, and Lovett [arXiv:1201.0330v2, Theorem 4.9], which is an arithmetic analogue of the strong regularity lemma for graphs developed by Alon, Fischer, Krivelevich, and Szegedy. Conlon and Fox [arXiv:1107.4829], as well as Kalyanasundaram and Shapira [arXiv:1107.4896v2], demonstrated that there are graphs for which any strong regularity partition must have size at least a wowzer-type function in the pseudorandomness parameter, and the primary aim of this paper is to match this bound in the setting of vector spaces over finite fields. The second kind of arithmetic regularity partition originates from higher-order arithmetic regularity lemmas. The upper bounds on the size of these partitions are known to be of tower-type growth. Previous work [arXiv:math/0310476v2, arXiv:1405.4409] demonstrated that this is unavoidable for the `linear' arithmetic regularity lemma of Green [arXiv:math/0310476v2], and the second contribution of this paper confirms that this continues to be necessary in the higher-order setting.