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3 solved out of 11 shown
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 $n$ \[\sum_{k\leq n}M_k > n^{1+c}?\]
The weaker conjecture that $\limsup M_n=\infty$ was proved by Wagner, who show that there is some $c>0$ with $M_n>(\log n)^c$ infinitely often.

This was solved by Beck [Be91], who proved that there exists some $c>0$ such that \[\max_{n\leq N} M_n > N^c.\]

Additional thanks to: Winston Heap
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 \[\int_0^{2\pi}\lvert f(\theta)\rvert \mathrm{d}\theta \leq 4.\]
Additional thanks to: Winston Heap
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?
More generally, if $A,B\subseteq \mathbb{R}$ are two countable dense sets then is there an entire function such that $f(A)=B$?
Additional thanks to: Boris Alexeev and Dustin Mixon
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 $0$.
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 $n$?
Originally a conjecture of Littlewood. The answer is yes (for all $n\geq 2$), proved by Balister, Bollobás, Morris, Sahasrabudhe, and Tiba [BBMST19].
Additional thanks to: Mehtaab Sawhney
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 \[f^{(k_n)}(z) = 0\textrm{ for all }z\in S_n?\]
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 \[\max_{\lvert z\rvert=1}\lvert P(z)\rvert \geq (1+c)\sqrt{n}?\]
The lower bound of $\sqrt{n}$ is trivial from Parseval's theorem. The answer is no (contrary to Erdős' initial guess). Kahane [Ka80] constructed 'ultraflat' polynomials $P(z)=\sum a_kz^k$ with $\lvert a_k\rvert=1$ such that \[P(z)=(1+o(1))\sqrt{n}\] uniformly forall $z\in\mathbb{C}$ with $\lvert z\rvert=1$, where the $o(1)$ term $\to 0$ as $n\to \infty$.

For more details see the paper [BoBo09] of Bombieri and Bourgain and, where Kahane's construction is improved to yield such a polynomial with \[P(z)=\sqrt{n}+O(n^{\frac{7}{18}}(\log n)^{O(1)})\] for all $z\in\mathbb{C}$ with $\lvert z\rvert=1$.

Additional thanks to: Mehtaab Sawhney
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 $N(k)$.
Erdős originally conjectured that $N(k)=2^k-1$, but this was disproved by Erdős and Bruijn. It is not even known whether $N(4)$ is finite.
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 \[f(n) \geq \exp(n^{c})?\]
Erdős and Szekeres [ErSz59] proved that $\lim f(n)^{1/n}=1$ and $f(n)>\sqrt{2n}$. Erdős proved an upper bound of $f(n) < \exp(n^{1-c})$ for some constant $c>0$ with probabilistic methods. Atkinson [At61] showed that $f(n) <\exp(cn^{1/2}\log n)$ for some constant $c>0$.
Additional thanks to: Zachary Chase
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 $k\to \infty$?
First investigated by Rényi and Rédei [Re47]. Erdős [Er49b] proved that $f(k)<k^{1-c}$ for some $c>0$. The conjecture that $f(k)\to \infty$ is due to Erdős and Rényi.
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 $A$?
A problem of Selfridge and Straus [SeSt58], who prove that this is true if $k=2$ and $\lvert A\rvert \neq 2^l$ (for $k\geq 0$).