Compact sets on the real line with large Hausdorff dimension contain curved three-point progressions for nonlinear functions such as polynomials with zero constant term and t^k log(1+t).
Mattila,Fourier analysis and Hausdorff dimension, Cambridge Studies in Advanced Mathematics, 150, Cambridge Univ
2 Pith papers cite this work. Polarity classification is still indexing.
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2026 2verdicts
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Extends the sharp dimensional threshold k from d=k+1 to the range k+1 ≤ d ≤ 2k and gives a non-trivial threshold of d-k for d > 2k in the simplex volume problem.
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A curved three-point pattern problem for fractal sets on the real line
Compact sets on the real line with large Hausdorff dimension contain curved three-point progressions for nonlinear functions such as polynomials with zero constant term and t^k log(1+t).
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On volumes of simplices in intermediate dimensions
Extends the sharp dimensional threshold k from d=k+1 to the range k+1 ≤ d ≤ 2k and gives a non-trivial threshold of d-k for d > 2k in the simplex volume problem.