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Fourier transforms of measures and algebraic relations on their supports. (English) Zbl 1179.42004
In an earlier paper [see Proc. Am. Math. Soc. 135, No. 12, 3823–3832 (2007; Zbl 1135.42002)], the author constructed a Borel measure \(\mu\) on the circle \(\mathbb T=\mathbb R/\mathbb Z\) such that for the natural \(q\) and for \(\varepsilon>0\) the Fourier coefficients \(\hat{\mu}(r)\) satisfy the relation \[ \hat{\mu}(r)=O\left(|r|^{\varepsilon-(2q)^{-1}}\right),\tag{1} \] but there do not exist \(x_j\in \text{supp }\mu\) and integers \(m_j\) satisfying some non trivial equation \(\sum^q_{j=1}m_jx_j=0\). Theorem 2.4 of this paper states that there exists a Borel measure \(\mu\) on the circle \(\mathbb T=\mathbb R/\mathbb Z\) such that the relation (1) is valid, but do not exist \(x_j\in \text{supp } \mu\) and integers \(m_j\) satisfying non trivial equation \(\sum^{q+1}_{j=1}m_jx_j=0\).
If \(\varepsilon>0\) is small, the set \(\left\{\sum^{q+1}_{j=1}x_j:x_j\in \text{supp }\mu\right\}\) has positive Lebesgue measure. It seems that the conclusion of the Theorem 2.4 is close to the best possible.
In Theorem 2.6 a closed set \(E\) is constructed such that the \(q\)-fold sum \(E+E+\dots+E\) has positive Lebesgue measure but there do not exist \(x_j\in \text{supp } \mu\) and integers \(m_j\) satisfying non trivial equation \(\sum^{2q-1}_{j=1}m_jx_j=0\).

MSC:
42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
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References:
[1] Kahane, J. P., Some random series of functions, 5, (1985), Cambridge University Press, Cambridge · Zbl 0571.60002
[2] Kaufman, R., A functional method for linear sets, Israel J. Math., 5, 185-187, (1967) · Zbl 0156.37403
[3] Kaufman, R., Small subsets of finite abelian groups, Annales de l’Institut Fourier, 18, 99-102, (1968) · Zbl 0175.30501
[4] Körner, T. W. K, Measures on independent sets, a quantitative version of a theorem of rudin, Proc. Amer. Math. Soc., 135, 12, 3823-3832, (2007) · Zbl 1135.42002
[5] Kuratowski, K., Topology, I, (1966), Academic Press, New York-London, Państwowe Wydawnictwo Naukowe, Warsaw · Zbl 0158.40802
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