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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
##### Keywords:
convolution; 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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