Bounds in a popular multidimensional nonlinear Roth theorem (Q6641592)

Szemerédi's theorem [\textit{E. Szemerédi}, Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)] asserts that any set of positive upper density in the integers contains arbitrarily long finite arithmetic progressions. \textit{V. Bergelson} and \textit{A. Leibman} [J. Am. Math. Soc. 9, No. 3, 725--753 (1996; Zbl 0870.11015)] extended this to nonlinear polynomial configurations and to higher dimensions. \textit{W. T. Gowers} [Geom. Funct. Anal. 8, No. 3, 529--551 (1998; Zbl 0907.11005); Geom. Funct. Anal. 11, No. 3, 465--588 (2001); Erratum 11, No. 4, 869 (2001; Zbl 1028.11005)] developed higher-order Fourier analysis to give quantitative density bounds in Szemerédi's theorem, initiating a broader programme that has been pushed very far by Peluse, Prendiville, and others [\textit{S. Peluse}, Forum Math. Pi 8, Paper No. e16, 55 p. (2020; Zbl 1522.11010); \textit{S. Peluse} and \textit{S. Prendiville}, Invent. Math. 238, No. 3, 865--903 (2024; Zbl 07953343); \textit{S. Peluse} and \textit{S. Prendiville}, Int. Math. Res. Not. 2022, No. 8, 5658--5684 (2022; Zbl 1508.11042)].\N\NNow Peluse, Prendiville, and Shao consider the two-dimensional, quadratic configuration \N\[ (x,y), (x+d,y), (x,y+d^2), \qquad \text{with} \qquad d \ne 0. \] \NThey show that if \(A \subset \{ 1,2, \ldots, N \}^2\) lacks this configuration then, for some absolute constant \(c > 0\), \[|A| \ll N^2/(\log N)^c.\] Moreover, they show that there's a popular difference: if \(N \ge e^{\varepsilon^{-C}}\) and \(|A| \ge \delta N^2\) then, for some \(d \ne 0\),\N\[\N\# \{ (x,y) \in A: (x+d, y), (x, y+d^2) \in A \} \ge (\delta^3 - \varepsilon) N^2.\N\]\NSimilarly, they show that if \(A \subset \{1,2,\ldots, N \}\) with \(N \ge e^{\varepsilon^{-C}}\) and \(|A| \ge \delta N\) then\N\[\N\# \{ x \in A: (x+d, x + d^2) \in A \} \ge (\delta^3 - \varepsilon) N.\N\]\NIn stark contrast to this, a tower-type relationship between \(N\) and \(\varepsilon\) is necessary and sufficient for the corresponding result for three-term arithmetic progressions [\textit{J. Fox} et al., J. Eur. Math. Soc. (JEMS) 25, No. 10, 3795--3831 (2023; Zbl 1533.11023)].\N\NThe essential new input is an inverse theorem for these configurations, with polynomial dependence. The authors speculate a potential future application to ergodic averages of the shape\N\[\NN^{-1} \sum_{d \le N} f(T_1^d x) g(T_2^{d^2}x),\N\]\Nwhere \(T_1, T_2\) are commuting measure-preserving transformations. In the case \(T_1 = T_2\), such a development was realised by Krause, Mirek, and Tao [\textit{B. Krause} et al., Ann. Math. (2) 195, No. 3, 997--1109 (2022; Zbl 1505.37010)].