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Finite configurations in sparse sets
, 2013
"... Let E ⊆ Rn be a closed set of Hausdorff dimension α. For m ≥ n, let {B1,..., Bk} be n × (m − n) matrices. We prove that if the system of matrices Bj is nondegenerate in a suitable sense, α is sufficiently close to n, and if E supports a probability measure obeying appropriate dimensionality and Fou ..."
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Let E ⊆ Rn be a closed set of Hausdorff dimension α. For m ≥ n, let {B1,..., Bk} be n × (m − n) matrices. We prove that if the system of matrices Bj is nondegenerate in a suitable sense, α is sufficiently close to n, and if E supports a probability measure obeying appropriate dimensionality and Fourier decay conditions, then for a range of m depending on n and k, the set E contains a translate of a nontrivial kpoint configuration {B1y,..., Bky}. As a consequence, we are able to establish existence of certain geometric configurations in Salem sets (such as parallelograms in R n and isosceles right triangles in R 2). This can be viewed as a multidimensional analogue of