Polynomial Roth theorems on sets of fractional dimensions
| Autor: | Robert B. Fraser, Shaoming Guo, Malabika Pramanik |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2019 |
| Předmět: |
Polynomial (hyperelastic model)
Degree (graph theory) Closed set General Mathematics 010102 general mathematics Dimension (graph theory) 01 natural sciences Fractal dimension Combinatorics symbols.namesake Fourier transform Mathematics - Classical Analysis and ODEs Hausdorff dimension 0103 physical sciences Classical Analysis and ODEs (math.CA) FOS: Mathematics symbols Mathematics - Combinatorics Combinatorics (math.CO) 010307 mathematical physics 0101 mathematics Probability measure Mathematics |
| Popis: | Let $E\subset \mathbb{R}$ be a closed set of Hausdorff dimension $\alpha\in (0, 1)$. Let $P: \mathbb{R}\to \mathbb{R}$ be a polynomial without a constant term whose degree is bigger than one. We prove that if $E$ supports a probability measure satisfying certain dimension condition and Fourier decay condition, then $E$ contains three points $x, x+t, x+P(t)$ for some $t>0$. Our result extends the one of Laba and the third author to the polynomial setting, under the same assumption. It also gives an affirmative answer to a question in Henriot, Laba and the third author. Comment: 16 pages |
| Databáze: | OpenAIRE |
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