Small-scale distribution of linear patterns of primes (Q6641563)

The complexity of a system of affine linear forms \(\Psi = (\psi_1,\hdots, \psi_t):\mathbb{Z}^d\rightarrow \mathbb{R}^t\) is the smallest integer \(s\geq 0\) such that \(\{\tilde{\psi_j}: [t]\backslash \{i\}\}\) can be partitioned into \(s+1\) classes such that \(\tilde{\psi_i}\) is not contained in the affine linear span of any of the classes, where \(\tilde{\psi}\) denotes the homogeneous part of \(\psi\). Letting \(\mathbf{1}_\mathcal{A}\) to be the characteristic function of \(\mathcal{A}\), the main result of the paper under review asserts that for a system \((\psi_1,\hdots, \psi_t):\mathbb{R}^d\rightarrow \mathbb{R}^t\) of finite complexity and for \(\lambda d>t\), there exist \(\delta_\lambda^+, \delta_\lambda^- > 0\) such that there exists \(X\) arbitrary large so that with \(H=(\log X)^{\lambda}\), there exist \(\mathbf{x}^\pm \in [X,2X]^d\) such that\N\[\N\sum_{\mathbf{x}\in \prod_{i=1}^d [x^+_i,x^+_i + H]} \prod_{i=1}^t\mathbf{1}_{\psi_i(\mathbf{x}) \textrm{ is prime}}\ge(1 + \delta_\lambda^+)\frac{H^d}{(\log X)^t}\prod_{p} \beta_p,\N\]\Nand\N\[\N\sum_{\mathbf{x}\in \prod_{i=1}^d [x^-_i,x^-_i + H]} \prod_{i=1}^t\mathbf{1}_{\psi_i(\mathbf{x}) \textrm{ is prime}}\le(1 - \delta_\lambda^-)\frac{H^d}{(\log X)^t}\prod_{p} \beta_p,\N\]\Nwhere the constant \(\beta_p\) is given by\N\[\N\beta_p = p^{-d} \sum_{\mathbf{x}\in \mathbb{F}_p^d} \prod_{i=1}^t \left(\frac{p}{p-1} \cdot \mathbf{1}_{(\psi_i(\mathbf{x}),p)=1}\right).\N\]\NA special case of the above result provides an answer in the negative to a Cramér-type model of a result due to Vinogradov asserting that for odd \(N\),\N\[\N\#\left\{p_1,p_2,p_3\in [0,N]: p_1+p_2+p_3 = N\right\} = \frac{\mathfrak{S}(N)N^2}{2\log^3 N},\N\]\Nwhere\N\[\N\mathfrak{S}(N)=\prod_{p | N}\bigg(1 - \frac{1}{(p - 1)^2}\bigg)\prod_{p\nmid N}\bigg(1 + \frac{1}{(p - 1)^3}\bigg)\asymp 1.\N\]\NMore precisely, the authors show that for all \(\lambda>3/2\) there exist \(\delta_\lambda^+, \delta_\lambda^- > 0\) such that for \(N\) a sufficiently large odd number satisfying \(N\ll X\le N/6\) with \(H =(\log X)^\lambda\) there exist \(x^\pm, y^\pm \in [X,2X]\) such that\N\[\N\#\left\{p_1 \in [x^\pm, x^\pm +H], p_2 \in [y^\pm, y^\pm +H]: N-p_1-p_2\text{ is prime}\right\}\geq\N(1\pm\delta_\lambda^\pm)\mathfrak{S}(N)\frac{H^2}{(\log X)^3}. \N\]