Weak hypergraph regularity and applications to geometric Ramsey theory

Let ? = 1 × …<!-- … <mml:mi>d ?<!-- ? <mml:msup> mathvariant="double-struck">R n \Delta =\Delta _1\times \ldots \times \Delta _d\subseteq \mathbb {R}^n , where alttext="double-struck n midline-horizontal-ellipsis Super d"> ?<!-- ? encoding="application/x-tex">\mathbb {R}^n=\mathbb {R}^{n_1}\times \cdots {R}^{n_d} with each i i"> i _i\subseteq {R}^{n_i} a non-degenerate simplex of alttext="n encoding="application/x-tex">n_i points. We prove that any set alttext="upper S S encoding="application/x-tex">S\subseteq plus + encoding="application/x-tex">n=n_1+\cdots +n_d positive Banach density necessarily contains an isometric copy all sufficiently large dilates the configuration Delta"> encoding="application/x-tex">\Delta . In particular such 2 2 {R}^{2d} alttext="d"> encoding="application/x-tex">d -dimensional cube side length alttext="lamda"> ?<!-- ? encoding="application/x-tex">\lambda for alttext="lamda greater-than-or-equal-to lamda 0 left-parenthesis right-parenthesis"> ?<!-- ? <mml:mn>0 ( stretchy="false">) encoding="application/x-tex">\lambda \geq \lambda _0(S) also analogous results underlying space being integer lattice. The proof is based on weak hypergraph regularity lemma and associated counting developed in context Euclidean spaces