Erdős · problems
OEIS index →← the campaignThe Erdős problems.
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#510If $A\subset \mathbb{Z}$ is a finite set of size $N$ then is there some absolute constant $c>0$ and $\theta$ such that\[\sum_{n\in A}\cos(n\theta) < -cN^{1/2}?\]Chowla's cosine problem · analysis · 1 Vela attempt (honest null) · AI tried (unverified) · N/A#1150Does there exist a constant $c>0$ such that, for all large $n$ and all polynomials $P$ of degree $n$ with coefficients $\pm 1$,\[\max_{\lvert z\rvert=1}\lvert P(z)\rvert > (1+c)\sqrt{n}?\]analysis · 1 Vela attempt (honest null) · N/A#119Let $z_i$ 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_…analysis · AI tried (unverified) · N/A#509Let $f(z)\in\mathbb{C}[z]$ be a monic non-constant polynomial. Can the set\[\{ z\in \mathbb{C} : \lvert f(z)\rvert \leq 1\}\]be covered by a set of circles the sum of whose radii is $\leq 2$?analysis · AI tried (unverified) · N/A#513Let $f=\sum_{n=0}^\infty a_nz^n$ be a transcendental entire function. What is the greatest possible value of\[\liminf_{r\to \infty} \frac{\max_n\lvert a_nr^n\rvert}{\max_{\lvert z\rvert=r}\lvert f(z)\…analysis · AI tried (unverified) · N/A#517Let $f(z)=\sum_{k=1}^\infty a_kz^{n_k}$ be an entire function (with $a_k\neq 0$ for all $k\geq 1$). Is it true that if $n_k/k\to \infty$ then $f(z)$ assumes every value infinitely often?analysis · AI tried (unverified) · N/A#522Let $f(z)=\sum_{0\leq k\leq n} \epsilon_k z^k$ be a random polynomial, where $\epsilon_k\in \{-1,1\}$ independently uniformly at random for $0\leq k\leq n$. Is it true that, if $R_n$ is the number of …analysis · AI tried (unverified) · N/A#906Is there an entire non-zero function $f:\mathbb{C}\to \mathbb{C}$ such that, for any infinite sequence $n_1<n_2<\cdots$, the set\[\{ z: f^{(n_k)}(z)=0 \textrm{ for some }k\geq 1\}\]is everywhe…analysis · AI tried (unverified) · N/A#973Does there exist a constant $C>1$ such that, for every $n\geq 2$, there exists a sequence $z_i\in \mathbb{C}$ with $z_1=1$ and $\lvert z_i\rvert \geq 1$ for all $1\leq i\leq n$ with\[\max_{2\leq k…analysis · AI tried (unverified) · N/A#996Let $n_1<n_2<\cdots$ be a lacunary sequence of integers, and let $f\in L^2([0,1])$. Let $f_n$ be the $n$th partial sum of the Fourier series of $f(x)$. Is there an absolute constant $C>0$ such that, i…analysis · AI tried (unverified) · N/A#997Call $x_1,x_2,\ldots \in (0,1)$ well-distributed if, for every $\epsilon>0$, if $k$ is sufficiently large then, for all $n>0$ and intervals $I\subseteq [0,1]$,\[\lvert \# \{ n<m\leq n+k : …analysis · solved · AI tried (unverified) · N/A#1002For any $0<\alpha<1$, let\[f(\alpha,n)=\frac{1}{\log n}\sum_{1\leq k\leq n}(\tfrac{1}{2}-\{ \alpha k\}).\]Does $f(\alpha,n)$ have an asymptotic distribution function?In other words, is there a…analysis · AI tried (unverified) · N/A#1038Determine the infimum and supremum of\[\lvert \{ x\in \mathbb{R} : \lvert f(x)\rvert < 1\}\rvert\]as $f\in \mathbb{R}[x]$ ranges over all non-constant monic polynomials, all of whose roots are rea…analysis · AI tried (unverified) · N/A#1041Let $f(z)=\prod_{i=1}^n(z-z_i)\in \mathbb{C}[z]$ with $\lvert z_i\rvert < 1$ for all $i$. Must there always exist a path of length less than $2$ in\[\{z: \lvert f(z)\rvert < 1\}\]which connect…analysis · AI tried (unverified) · N/A#256Let $n\geq 1$ and $f(n)$ be maximal such that for any integers $1\leq a_1\leq \cdots \leq a_n$ we have\[\max_{\lvert z\rvert=1}\left\lvert \prod_{i}(1-z^{a_i})\right\rvert\geq f(n).\]Estimate $f(n)$ -…analysis · AI tried (unverified) · N/A#514Let $f(z)$ be an entire transcendental function. Does there exist a path $L$ so that, for every $n$,\[\lvert f(z)/z^n\rvert \to \infty\]as $z\to \infty$ along $L$?Can the length of this path be estima…analysis · AI tried (unverified) · N/A#521Let $(\epsilon_k)_{k\geq 0}$ be independently uniformly chosen at random from $\{-1,1\}$. If $R_n$ counts the number of real roots of $f_n(z)=\sum_{0\leq k\leq n}\epsilon_k z^k$ then is it true that, …analysis · AI tried (unverified) · N/A#524For any $t\in (0,1)$ let $t=\sum_{k=1}^\infty \epsilon_k(t)2^{-k}$ (where $\epsilon_k(t)\in \{0,1\}$). What is the correct order of magnitude (for almost all $t\in(0,1)$) for\[M_n(t)=\max_{x\in [-1,1]…analysis · AI tried (unverified) · N/A#671Given $a_{i}^n\in [-1,1]$ for all $1\leq i\leq n<\infty$ we define $p_{i}^n$ as the unique polynomial of degree $n-1$ such that $p_{i}^n(a_{i}^n)=1$ and $p_{i}^n(a_{i'}^n)=0$ if $1\leq i'\leq n$ w…analysis · AI tried (unverified) · N/A#987Let $x_1,x_2,\ldots \in (0,1)$ be an infinite sequence and let\[A_k=\limsup_{n\to \infty}\left\lvert \sum_{j\leq n} e(kx_j)\right\rvert,\]where $e(x)=e^{2\pi ix}$.Is it true that\[\limsup_{k\to \infty…analysis · solved · AI tried (unverified) · N/A#990Let $f=a_0+\cdots+a_dx^d\in \mathbb{C}[x]$ be a polynomial. Is it true that, if $f$ has roots $z_1,\ldots,z_d$ with corresponding arguments $\theta_1,\ldots,\theta_d\in [0,2\pi]$, then for all interva…analysis · solved · AI tried (unverified) · N/A#995Let $n_1<n_2<\cdots$ be a lacunary sequence of integers and $f\in L^2([0,1])$. Estimate the growth of, for almost all $\alpha$,\[\sum_{1\leq k\leq N}f(\{ \alpha n_k\}).\]For example, is it tru…analysis · AI tried (unverified) · N/A#1039Let $f(z)=\prod_{i=1}^n(z-z_i)\in \mathbb{C}[z]$ with $\lvert z_i\rvert \leq 1$ for all $i$. Let $\rho(f)$ be the radius of the largest disc which is contained in $\{z: \lvert f(z)\rvert< 1\}$. De…analysis · AI tried (unverified) · N/A#1040Let $F\subseteq \mathbb{C}$ be a closed infinite set, and let $\mu(F)$ be the infimum of\[\lvert \{ z: \lvert f(z)\rvert < 1\}\rvert,\]as $f$ ranges over all polynomials of the shape $\prod (z-z_i…analysis · AI tried (unverified) · N/A#1044Let $f(z)=\prod_{i=1}^n(z-z_i)\in\mathbb{C}[x]$ where $\lvert z_i\rvert\leq 1$ for all $i$. If $\Lambda(f)$ is the maximum of the lengths of the boundaries of the connected components of\[\{ z: \lvert…analysis · solved · AI tried (unverified) · N/A#1045Let $z_1,\ldots,z_n\in \mathbb{C}$ with $\lvert z_i-z_j\rvert\leq 2$ for all $i,j$, and\[\Delta(z_1,\ldots,z_n)=\prod_{i\neq j}\lvert z_i-z_j\rvert.\]What is the maximum possible value of $\Delta$? Is…analysis · AI tried (unverified) · N/A#1117Let $f(z)$ be an entire function which is not a monomial. Let $\nu(r)$ count the number of $z$ with $\lvert z\rvert=r$ such that $\lvert f(z)\rvert=\max_{\lvert z\rvert=r}\lvert f(z)\rvert$. (This is …analysis · AI tried (unverified) · N/A#1119Let $\mathfrak{m}$ be an infinite cardinal with $\aleph_0<\mathfrak{m}<\mathfrak{c}=2^{\aleph_0}$. Let $\{f_\alpha\}$ be a family of entire functions such that, for every $z_0\in \mathbb{C}$, …analysis · solved · AI tried (unverified) · N/A#1120Let $f\in \mathbb{C}[z]$ be a monic polynomial of degree $n$, all of whose roots satisfy $\lvert z\rvert\leq 1$. Let\[E= \{ z : \lvert f(z)\rvert \leq 1\}.\]What is the shortest length of a path in $E…analysis · AI tried (unverified) · N/A#1129For $x_1,\ldots,x_n\in [-1,1]$ let\[l_k(x)=\frac{\prod_{i\neq k}(x-x_i)}{\prod_{i\neq k}(x_k-x_i)},\]which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\neq k$.Describe which choice of $x_i$ mini…analysis · solved · AI tried (unverified) · N/A#1130For $x_1,\ldots,x_n\in [-1,1]$ let\[l_k(x)=\frac{\prod_{i\neq k}(x-x_i)}{\prod_{i\neq k}(x_k-x_i)},\]which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\neq k$.Let $x_0=-1$ and $x_{n+1}=1$ and\[\…analysis · solved · AI tried (unverified) · N/A#1131For $x_1,\ldots,x_n\in [-1,1]$ let\[l_k(x)=\frac{\prod_{i\neq k}(x-x_i)}{\prod_{i\neq k}(x_k-x_i)},\]which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\neq k$.What is the minimal value of\[I(x_1…analysis · AI tried (unverified) · N/A#1132For $x_1,\ldots,x_n\in [-1,1]$ let\[l_k(x)=\frac{\prod_{i\neq k}(x-x_i)}{\prod_{i\neq k}(x_k-x_i)},\]which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\neq k$.Let $x_1,x_2,\ldots\in [-1,1]$ be a…analysis · AI tried (unverified) · N/A#1133Let $C>0$. There exists $\epsilon>0$ such that if $n$ is sufficiently large the following holds.For any $x_1,\ldots,x_n\in [-1,1]$ there exist $y_1,\ldots,y_n\in [-1,1]$ such that, if $P$ is a…analysis · AI tried (unverified) · N/A#228Does 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 c…analysis · solved · N/A#229Let $(S_n)_{n\geq 1}$ be a sequence of sets of complex numbers, none of which have a finite limit point. Does there exist an entire transcendental function $f(z)$ such that, for all $n\geq 1$, there e…analysis · solved · N/A#494If $A\subset \mathbb{C}$ is a finite set and $k\geq 1$ then let\[A_k = \{ z_1+\cdots+z_k : z_i\in A\textrm{ distinct}\}.\]For $k>2$ does the multiset $A_k$ (together with the size of $A$) uniquely…analysis · solved · N/A#516Let $f(z)=\sum_{k\geq 1}a_k z^{n_k}$ be an entire function of finite order such that $\lim n_k/k=\infty$. Let $M(r)=\max_{\lvert z\rvert=r}\lvert f(z)\rvert$ and $m(r)=\min_{\lvert z\rvert=r}\lvert f(…analysis · solved · N/A#1043Let $f\in \mathbb{C}[x]$ be a monic non-constant polynomial. Must there exist a straight line $\ell$ such that the projection of\[\{ z: \lvert f(z)\rvert\leq 1\}\]onto $\ell$ has measure at most $2$?analysis · solved · N/A#225Let\[ f(\theta) = \sum_{0\leq k\leq n}c_k e^{ik\theta}\]be a trigonometric polynomial all of whose roots are real, such that $\max_{\theta\in [0,2\pi]}\lvert f(\theta)\rvert=1$. Then\[\int_0^{2\pi}\lv…analysis · solved · N/A#226Is 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?analysis · solved · N/A#227Let $f=\sum_{n=0}^\infty a_nz^n$ be an entire function which is not a polynomial. Is it true that if\[\lim_{r\to \infty} \frac{\max_n\lvert a_nr^n\rvert}{\max_{\lvert z\rvert=r}\lvert f(z)\rvert}\]exi…analysis · solved · N/A#230Let $P(z)=\sum_{1\leq k\leq n}a_kz^k$ 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 …analysis · solved · N/A#395If $z_1,\ldots,z_n\in \mathbb{C}$ with $\lvert z_i\rvert=1$ then is it true that the probability that\[\lvert \epsilon_1z_1+\cdots+\epsilon_nz_n\rvert \leq \sqrt{2},\]where $\epsilon_i\in \{-1,1\}$ un…reverse Littlewood-Offord problem · analysis · solved · N/A#485Let $f(k)$ 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 $k\to \infty$?analysis · solved · possible#511Let $f(z)\in \mathbb{C}[z]$ be a monic polynomial of degree $n$. Is it true that, for every $c>1$, the set\[\{ z\in \mathbb{C} : \lvert f(z)\rvert< 1\}\]has at most $O_c(1)$ many connected compone…analysis · solved · N/A#512Is it true that, if $A\subset \mathbb{Z}$ is a finite set of size $N$, then\[\int_0^1 \left\lvert \sum_{n\in A}e(n\theta)\right\rvert \mathrm{d}\theta \gg \log N,\]where $e(x)=e^{2\pi ix }$?Littlewood's conjecture · analysis · solved · N/A#515Let $f(z)$ be an entire function, not a polynomial. Does there exist a locally rectifiable path $C$ tending to infinity such that, for every $\lambda>0$, the integral\[\int_C \lvert f(z)\rvert^{-\…analysis · solved · N/A#519Let $z_1,\ldots,z_n\in \mathbb{C}$ with $z_1=1$. Must there exist an absolute constant $c>0$ such that\[\max_{1\leq k\leq n}\left\lvert \sum_{i}z_i^k\right\rvert>c?\]analysis · solved · N/A#523Let $f(z)=\sum_{0\leq k\leq n} \epsilon_k z^k$ be a random polynomial, where $\epsilon_k\in \{-1,1\}$ independently uniformly at random for $0\leq k\leq n$. Does there exist some constant $C>0$ su…analysis · solved · N/A#525Is it true that all except at most $o(2^n)$ many degree $n$ polynomials with $\pm 1$-valued coefficients $f(z)$ have $\lvert f(z)\rvert <1$ for some $\lvert z\rvert=1$?What is the behaviour of\[m(…analysis · solved · possible#527Let $a_n\in \mathbb{R}$ be such that $\sum_n \lvert a_n\rvert^2=\infty$ and $\lvert a_n\rvert=o(1/\sqrt{n})$. Is it true that, for almost all $\epsilon_n=\pm 1$, there exists some $z$ with $\lvert z\r…analysis · solved · N/A#907Let $f:\mathbb{R}\to \mathbb{R}$ be such that $f(x+h)-f(x)$ is continuous for every $h>0$. Is it true that\[f=g+h\]for some continuous $g$ and additive $h$ (i.e. $h(x+y)=h(x)+h(y)$)?analysis · solved · N/A#908Let $f:\mathbb{R}\to \mathbb{R}$ be such that $f(x+h)-f(x)$ is measurable for every $h>0$. Is it true that\[f=g+h+r\]where $g$ is continuous, $h$ is additive (so $h(x+y)=h(x)+h(y)$), and $r(x+h)-r…analysis · solved · N/A#909Let $n\geq 2$. Is there a space $S$ of dimension $n$ such that $S^2$ also has dimension $n$?analysis · solved · N/A#974Let $z_1,\ldots,z_n\in \mathbb{C}$ be a sequence such that $z_1=1$. Suppose that the sequence of\[s_k=\sum_{1\leq i\leq n}z_i^k\]contains infinitely many $(n-1)$-tuples of consecutive values of $s_k$ …analysis · solved · N/A#994Let $E\subseteq (0,1)$ be a meaurable subset with Lebesgue measure $\lambda(E)$. Is it true that, for almost all $\alpha$,\[\lim_{n\to \infty}\frac{1}{n}\sum_{1\leq k\leq n}1_{\{k\alpha \}\in E}=\lamb…analysis · solved · N/A#998Let $\alpha$ be an irrational number. Is it true that if, for all large $n$,\[\#\{ 1\leq m\leq n : \{ \alpha m\} \in [u,v)\} = n(v-u)+O(1)\]then $u=\{\alpha k\}$ and $v=\{\alpha \ell\}$ for some integ…analysis · solved · N/A#1042Let $F\subset\mathbb{C}$ be a closed set of transfinite diameter $1$ which is not contained in any closed disc of radius $1$. If $f(z)=\prod_{i=1}^n(z-z_i)\in\mathbb{C}[x]$ with all $z_i\in F$ then ca…analysis · solved · N/A#1046Let $f\in \mathbb{C}[x]$ be a monic polynomial and\[E=\{ z: \lvert f(z)\rvert <1\}.\]If $E$ is connected then is $E$ contained in a disc of radius $2$?analysis · solved · N/A#1047Let $f\in \mathbb{C}[x]$ be a monic polynomial with $m$ distinct roots, and let $c>0$ be a constant small enough such that $\{ z: \lvert f(z)\rvert\leq c\}$ has $m$ distinct connected components.M…analysis · solved · N/A#1048If $f\in \mathbb{C}[x]$ is a monic polynomial with all roots satisfying $\lvert z\rvert \leq r$ for some $r<2$, then must\[\{ z: \lvert f(z)\rvert <1\}\]have a connected component with diameter $>2-r$…analysis · solved · N/A#1114Let $f(x)\in \mathbb{R}[x]$ be a polynomial of degree $n$ whose roots $\{a_0<\cdots<a_n\}$ are all real and form an arithmetic progression. The differences between consecutive zeros of $f'(x)$…analysis · solved · N/A#1115Let $f(z)$ be an entire function of finite order, and let $\Gamma$ be a rectifiable path on which $f(z)\to \infty$. Let $\ell(r)$ be the length of $\Gamma$ in the disc $\lvert z\rvert<r$. Find a p…analysis · solved · N/A#1116For a meromorphic function $f$ let $n(r,a)$ count the number of roots of $f(z)=a$ in the disc $\lvert z\rvert <r$. Does there exist a meromorphic (or entire) $f$ such that for every $a\neq b$\[\li…analysis · solved · N/A#1118Let $f(z)$ be a non-constant entire function such that, for some $c$, the set $E(c)=\{ z: \lvert f(z)\rvert >c\}$ has finite measure. What is the minimum growth rate of $f(z)$?If $E(c)$ has finite…analysis · solved · N/A#1125Let $f:\mathbb{R}\to \mathbb{R}$ be such that\[2f(x) \leq f(x+h)+f(x+2h)\]for every $x\in \mathbb{R}$ and $h>0$. Must $f$ be monotonic?analysis · solved · N/A#1126If\[f(x+y)=f(x)+f(y)\]for almost all $x,y\in \mathbb{R}$ then there exists a function $g$ such that\[g(x+y)=g(x)+g(y)\]for all $x,y\in\mathbb{R}$ such that $f(x)=g(x)$ for almost all $x$.analysis · solved · N/A#1151Given $a_1,\ldots,a_n\in [-1,1]$ let\[\mathcal{L}^nf(x) = \sum_{1\leq i\leq n}f(a_i)\ell_i(x)\]be the unique polynomial of degree $n-1$ which agrees with $f$ on $a_i$ for $1\leq i\leq n$ (that is, the…analysis · N/A#1152For $n\geq 1$ fix some sequence of $n$ distinct numbers $x_{1n},\ldots,x_{nn}\in [-1,1]$. Let $\epsilon=\epsilon(n)\to 0$. Does there always exist a continuous function $f:[-1,1]\to \mathbb{R}$ such t…analysis · N/A#1153For $x_1,\ldots,x_n\in [-1,1]$ let\[l_k(x)=\frac{\prod_{i\neq k}(x-x_i)}{\prod_{i\neq k}(x_k-x_i)},\]which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\neq k$.Let\[\lambda(x)=\sum_k \lvert l_k(x…analysis · solved · N/A#1154Does there exist, for every $\alpha \in [0,1]$, a ring or field in $\mathbb{R}$ with Hausdorff dimension $\alpha$?analysis · solved · N/A#1197Let $E\subset (0,\infty)$ be a set of positive measure. Is it true that, for almost all $x>0$, for all sufficiently large (depending on $x$) integers $n$ there exists an integer $r\geq 1$ such tha…analysis · solved · N/A#1215Does there exist a constant $C$ such that for every polynomial $P$ with $P(0)=1$, all of whose roots are on the unit circle, there exists a path in\[\{ z: \lvert P(z)\rvert < 1\}\]which connects $…analysis · solved · N/A