Finite configurations in sparse sets (Q279766)

The paper deals with the problem of finding finite geometric configurations in sparse multidimensional sets. The main result is the following: If \(n\lceil (k+1)/2\rceil \leq m < nk, 2(nk-m)/k<\beta<n\), if a system of matrices \(\{B_1,\dots,B_k\}\) of the dimension \(n\times (n-m)\) is non-degenerate in a suitable sense and if a set \(E\subset\mathbb{R}^n\) with \(|E|=0\) supports a positive, finite, Radon measure \(\mu\) with the two conditions: (a) (Ball condition): \(\sup_{x\in E,\; 0<r<1} \mu(B(x;r))/r^\alpha \leq C\) for some \(\alpha\in (n-\varepsilon_0,n)\); (b) (Fourier decay): \(\sup_{\xi\in \mathbb{R}^n} \widehat{\mu}(\xi)(1+|\xi|)^{\beta/2}\leq C\), then assuming that \(\varepsilon_0\) is sufficiently small number the set \(E\) contains a translate of a non-trivial \(k\)-point configuration \(\{B_1y,\dots,B_ky\}\). As a consequences there are obtained the following results: 1) Let \(a,b,c\) be three distinct points on the plane. If \(\beta>4/3\) and a set \(E\subset\mathbb{R}^2\) with \(|E|=0\) supports a positive, finite, Radon measure \(\mu\) with the ball and Fourier decay conditions, then assuming that \(\varepsilon_0\) is sufficiently small (depending on \(C, \beta\) and on the configuration \(\{a,b,c\}\)) the set \(E\) contains a triangle \(\Delta xyz\) similar to the triangle \(\Delta abc\); 2) Let \(a,b,c\) be three distinct collinear points in \(\mathbb{R}^n\). If \(\beta>2n/3\) and a set \(E\subset\mathbb{R}^n\) with \(|E|=0\) supports a positive, finite, Radon measure \(\mu\) with the ball and Fourier decay conditions, then assuming that \(\varepsilon_0\) is sufficiently small (depending on \(C, \beta\) and on the configuration \(\{a,b,c\}\)) the set \(E\) contain three distinct points \(x,y,z\) that form a similar image of the triple \(a,b,c\); 3) If \(\beta>n/2\) and a set \(E\subset\mathbb{R}^n\) with \(|E|=0\) supports a positive, finite, Radon measure \(\mu\) with the ball and Fourier decay conditions, then assuming that \(\varepsilon_0\) is sufficiently small (depending on \(C\) and \(\beta\)) the set \(E\) contains a parallelogram \(\{x, x+y, x+z, x+y+z\}\), where the four points are distinct. The paper continues the previous work of the last two authors concerning 3-term arithmetic progressions in one-dimensional sparse sets: [Geom. Funct. Anal. 19, No. 2, 429--456 (2009; Zbl 1184.28010)].