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Measures on independent sets, a quantitative version of Rudin’s theorem. (English) Zbl 1135.42002

A closed set \(E\) is called independent if, given distinct points \(x_1,x_2,\dots,x_q\in E\), the only solution to the equation
\[ \sum_{j=1}^q m_jx_j=0 \]
with \(m_j\in\mathbb Z\), is the trivial solution \(m_1=m_2=\cdots=m_q=0\).
The author works on the circle \(\mathbb T=\mathbb R/\mathbb Z\) and constructs measures with independent support whose Fourier coefficients decrease as fast as possible.
The main result of the author is given in Theorem 1.4 which is a modified version of Rudin’s theorem stated in Theorem 1.3. These results are stated as follows:
Theorem 1.3 (Rudin’s theorem). There exists a probability measure \(\mu\) such that \(\widehat{\mu}(r)\to 0\) as \(|r|\to\infty\) but support of \(\mu\) is independent.
The author proves the following version of Rudin’s theorem.
Theorem 1.4. Suppose that \(\varphi:N\to\mathbb R\) is a sequence of positive numbers with \(r^\alpha\varphi(r)\to\infty\) as \(r\to\infty\) whenever \(\alpha>0\). Then there exists a probability measure \(\mu\) such that \(\varphi(|r|)\geq|\widehat{\mu}(r)|\) for all \(r\neq 0\), but support of \(\mu\) is independent.
Numerous other theorems and lemmas are proved.

MSC:

42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
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