A famous classical theorem of Cantor says that if a trigonometric series converges to zero everywhere, then all the coefficients are zero. This led him to the concept of a set of uniqueness. A closed set is a set of uniqueness if it does not support any distribution whose Fourier coefficients vanish at infinity. Lebesgue showed that a set of uniqueness is necessarily of Lebesgue measure zero. But the converse is not true, since there are probability measures supported on sets of measure zero with Fourier coefficients \(O(\log |n|)^{-1/2}\), as was shown by Mensǒv. There are classical refinements of the last result due to Littlewood and Wiener-Wintner. A remarkable result of Ivašev-Musatov (1950’s) showed that there are probability measures \(\mu\) supported on sets of measure zero with \(|\hat{\mu}(n)| \leq \phi(n), n\neq 0,\) for any ‘well-behaved’ sequence \(\phi(n)\geq 0\) with \(\Sigma \phi(n)^2 = \infty\). This paper is concerned with the question: if the support of a distribution has Hausdorff h-measure zero, how fast can its Fourier coefficients decay? The author, well-known for his delicate and deep constructions in Fourier analysis, proves the following result:
Assuming appropriate good behaviour of \(h\) and \(\phi\), if \(\Sigma \phi(n)^2 = \infty\), then there is a non-zero distribution \(S\) supported on a set of zero Hausdorff h-measure such that \[ |\hat{S}(n)|\leq \phi(|n|)(|n|h(|n|^{-1}))^{1/2}(\log (|n|\max \{h(|n|^{-1}),\phi(n)^2\}))^{1/2}, n\neq 0 \] .
Key new ideas involve certain probabilistic constructions on finite abelian groups and then these are carried over to the setting of the unit circle \(\mathbb{T}\). These and arguments of Ivašev-Musatov yield smooth functions on the circle with suitable growth conditions on their Fourier coefficients and this result contains the essence of the main theorem. The proof is completed by an ingenious application of the Baire category argument.