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Creators/Authors contains: "Lyall, Neil"

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  1. ; (, Analysis & PDE)
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  2. ; ; (, American Journal of Mathematics)
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  3. ; (, Mathematical Proceedings of the Cambridge Philosophical Society)
    Abstract We establish that any subset of ℝ d of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates of any fixed two-dimensional rectangle provided d ⩾ 4. We further present an extension of this result to configurations that are the product of two non-degenerate simplices; specifically we show that if Δ k 1 and Δ k 2 are two fixed non-degenerate simplices of k 1 + 1 and k 2 + 1 points respectively, then any subset of ℝ d of positive upper Banach density with d ⩾ k 1 + k 2 + 6 will necessarily contain an isometric copy of all sufficiently large dilates of Δ k 1 × Δ k 2 . A new direct proof of the fact that any subset of ℝ d of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates of any fixed non-degenerate simplex of k + 1 points provided d ⩾ k + 1, a result originally due to Bourgain, is also presented. 
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  4. ; ; (, Proceedings of the American Mathematical Society)
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