$100

Let $z_i$ be an infinite sequence of complex numbers such that $\lvert z_i\rvert=1$ for all $i\geq 1$, and for $n\geq 1$ let \[p_n(z)=\prod_{i\leq n} (z-z_i).\] Let $M_n=\max_{\lvert z\rvert=1}\lvert p_n(z)\rvert$. Is it true that $\limsup M_n=\infty$? Is it true that there exists $c>0$ such that for infinitely many $n$ we have $M_n > n^c$, or even that for all $n$ \[\sum_{k\leq n}M_k > n^{1+c}?\]

#119: [Er57][Er61][Er90]

analysis, polynomials

Let \[ f(\theta) = \sum_{k\geq 1}c_k e^{ik\theta}\] be a trigonometric polynomial (so that the $c_k\in \mathbb{C}$ are finitely supported) with real roots such that $\max_{\theta\in [0,2\pi]}\lvert f(\theta)\rvert=1$. Then \[\int_0^{2\pi}\lvert f(\theta)\rvert \mathrm{d}\theta \leq 4.\]

#225: [Er57]

Is there an entire non-linear function $f$ such that, for all $x\in\mathbb{R}$, $x$ is rational if and only if $f(x)$ is?

#226: [Er57]

If $f(z)=\sum_{n\geq 0}a_nz^n$ is an entire function such that \[\lim_{r\to \infty}\frac{\max_n \lvert a_n\rvert r^n}{\max_\theta \lvert f(re^{i\theta})\rvert}\] exists then this limit must be $0$.

#227: [Er57]

Does there exist, for all large $n$, a polynomial $P$ of degree $n$, with coefficients $\pm 1$,, such that \[\sqrt{n} \ll \lvert P(z) \rvert \ll \sqrt{n}\] for all $\lvert z\rvert =1$, with the implied constants independent of $z$ and $n$?

#228: [Er57]

analysis, polynomials

Let $(S_n)_{n\geq 1}$ be a sequence of sets, none of which has a finite limit point. Does there exist an entire function $f(z)$ such that, for all $n\geq 1$, there exists some $k_n\geq 0$ such that \[f^{(k_n)}(z) = 0\textrm{ for all }z\in S_n?\]

#229: [Er57]

Let $P(z)=\sum_{1\leq k\leq n}a_kz^k$ for some $a_k\in \mathbb{C}$ with $\lvert a_k\rvert=1$ for $1\leq k\leq n$. Does there exist a constant $c>0$ such that, for $n\geq 2$, we have \[\max_{\lvert z\rvert=1}\lvert P(z)\rvert \geq (1+c)\sqrt{n}?\]

#230: [Er57]

analysis, polynomials

Let $k\geq 1$ and define $N(k)$ to be the minimal $N$ such that any string $s\in \{1,\ldots,k\}^N$ contains two adjacent blocks such that each is a rearrangement of the other. Estimate $N(k)$.

#232: [Er57]

Let $n\geq 1$ and $f(n)$ be maximal such that, for every set $A\subset \mathbb{N}$ with $\lvert A\rvert=n$, we have \[\max_{\lvert z\rvert=1}\left\lvert \prod_{n\in A}(1-z^n)\right\rvert\geq f(n).\] Estimate $f(n)$ - in particular, is it true that there exists some constant $c>0$ such that \[f(n) \geq \exp(n^{c})?\]

#256: [Er61]

Let $f(k)$ be the minimum number of terms in $P(x)^2$, where $P\in \mathbb{Q}[x]$ ranges over all polynomials with exactly $k$ non-zero terms. Is it true that $f(k)\to\infty$ as $k\to \infty$?

#485: [Er61]

analysis, polynomials

Let $A\subset \mathbb{C}$ be a finite set, for any $k\geq 1$ let \[A_k = \{ z_1\cdots z_k : z_i\in A\textrm{ distinct}\}.\] For $k>2$ does the set $A_k$ uniquely determine the set $A$?

#494: [Er61]

analysis, additive combinatorics