Fourier transforms of measures and algebraic relations on their supports (Q734596)

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 \(\sum^{2q-1}_{j=1}m_jx_j=0\).