 3 solved out of 11 shown
$100 Let$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}?$ 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\$).