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Source. Remark 4.1, p. 346 (section 4), with the Lemma stated there, of A. Dvoretzky and P. Erdős, Divergence of random power series, Michigan Math. J. 6 (1959), 343--347, the edition named on the source card.

Read depth. Claims checked: the remark and the Lemma were read clause by clause on the page image of p. 346. The paper proves neither: it announces the construction ("we can, however, show") and names the Lemma as its main new tool, with no proof printed in this paper. Nothing here is independently reviewed.

Statement

Setting as on the Theorem's page: (2) is the condition $\sum_{n=0}^{\infty}|a_n|^2=\infty$, (3) the Theorem's limsup condition, and $\mathscr F\{a_n\}$ the family of Rademacher-signed power series $\sum\phi_n(t)a_nz^n$.

Remark 4.1 (p. 346). The authors do not know whether condition (3) is best possible. They assert that (3) cannot be replaced by (2): there is a monotone sequence $\{a_n\}$ satisfying (2) such that almost all series of $\mathscr F\{a_n\}$ have, on every arc of $|z|=1$, a set of points of convergence of the power of the continuum.

Lemma (p. 346). For every $\alpha<\beta$ and every $\varepsilon>0$, only $o(2^n)$ of the $2^n$ choices of signs $\pm$ satisfy

$$\min_{\alpha\le t\le\beta}\ \max_{1\le m\le n}\ \Bigl|\sum_{j=1}^{m}\pm e^{2\pi ijt}\Bigr|>\varepsilon\sqrt n .$$

Proof pointer

None in this paper: the construction and the Lemma are stated only.

Dependencies

None stated.

Bears on

• Problem 527: the problem asks whether, for real $a_n$ with $\sum|a_n|^2=\infty$ and $|a_n|=o(1/\sqrt n)$, almost every choice of signs gives a series that converges at some point of $|z|=1$. Remark 4.1 asserts, without proof here, that for some monotone sequence with $\sum|a_n|^2=\infty$ almost every choice of signs gives a series with, on every arc of $|z|=1$, a set of convergence points of the power of the continuum. The remark does not state how the sequence compares with $1/\sqrt n$, and the paper does not pose the problem.