Continuous measures with large partial sums (Q2783043)

It is known that if \(f\in L^1({\mathbb T})\) then the Fourier partial sums \(S_Nf=\sum_{-N}^N\widehat f(n)e^{-inx}\) satisfy the growth condition \(\|S_Nf\|=o(\log N)\), where \(\|\cdot\|\) denotes the \(L^1\)-norm on \(\mathbb T\), but if \(\mu\) is a measure on \(\mathbb T\) then we only have \(\|S_N\mu\|=O(\log n)\). Moreover, it is known that if \(\liminf_{N\to\infty}\|S_N\mu\|/\log N=0\) then \(\mu\) is a continuous measure on \(\mathbb T\). If \(\mu\) is discrete then \textit{M. E. Andersson} [Spaces with uniformly norm-bounded partial sums. II, Res. Rep. Math. No. 11, Dep. Math., Stockholm Univ., Stockholm, 1999] proved the relation NEWLINE\[NEWLINE \lim_{N\to\infty}\frac{\|S_N\mu\|}{\log N}=\frac 4{\pi^2}\|\mu\|, NEWLINE\]NEWLINE where \(\|\mu\|\) denotes the total variation of \(\mu\). The main result of the present paper is the theorem that if \(K\) is a sufficiently singular Cantor set, then there exists a sequence \(N_n\to\infty\) of positive integers (depending only on \(K\)) such that the above-mentioned relation is fulfilled along the sequence \(N_n\) for every measure \(\mu\) supported by \(K\).