Claims to confirm the L^α-Littlewood, L^1-Newman, and L^∞-Erdős conjectures via a short Clarkson-inequality argument, but the key norm-convergence step is unjustified.
The sequence of partial sums of a unimodular power series is not ultraflat
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abstract
We show that if $(a_j)_{j=0}^\infty$ is a sequence of numbers $a_j \in {\Bbb C}$ with $|a_j|=1$, and $$P_n(z) = \sum_{j=0}^n{a_jz^j}\,, \qquad n=0,1,2,\ldots\,,$$ then $(P_n)$ is NOT an ultraflat sequence of unimodular polynomials. This answers a question raised by Zachary Chase.
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A generalization of Littlewood's $L^\alpha$ flat theorem, $\alpha>0$
Claims to confirm the L^α-Littlewood, L^1-Newman, and L^∞-Erdős conjectures via a short Clarkson-inequality argument, but the key norm-convergence step is unjustified.