A problem of Erdős, Herzog, and Piranian [EHP58].

The lower bound is easy: this is $\geq n$ and equality holds if and only if $p(z)=z^n$. The assumption that the set is connected is necessary, as witnessed for example by $p(z)=z^2+10z+1$.

The Chebyshev polynomials show that $n^2/2$ is best possible here. Erdős originally conjectured this without the $o(1)$ term but Szabados observed that was too strong. Pommerenke [Po59a] proved an upper bound of $\frac{e}{2}n^2$.

Eremenko and Lempert [ErLe94] have shown this is true, and in fact Chebyshev polynomials are the extreme examples.

Conjectured by Erdős, Herzog, and Piranian [ErHePi58]. The lower bound $\mu(A) \gg n^{-4}$ follows from a result of Pommerenke [Po61]. The stronger lower bound $\gg (\log n)^{-O(1)}$ is still open.

Wagner [Wa88] proves, for $n\geq 3$, the existence of such polynomials with \[\mu(A) \ll_\epsilon (\log\log n)^{-1/2+\epsilon}\] for all $\epsilon>0$.

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 \[\max_{n\leq N} M_n > N^c.\]

This was solved independently by Kristiansen [Kr74] (only in the case when $c_k\in\mathbb{R}$) and Saff and Sheil-Small [SSS73] (for general $c_k\in \mathbb{C}$).

More generally, if $A,B\subseteq \mathbb{R}$ are two countable dense sets then is there an entire function such that $f(A)=B$?

Solved by Barth and Schneider [BaSc70], who proved that if $A,B\subset\mathbb{R}$ are countable dense sets then there exists a transcendental entire function $f$ such that $f(z)\in B$ if and only if $z\in A$. In [BaSc71] they proved the same result for countable dense subsets of $\mathbb{C}$.

Clunie (unpublished) proved this if $a_n\geq 0$ for all $n$. This was disproved in general by Clunie and Hayman [ClHa64], who showed that the limit can take any value in $[0,1/2]$.

See also [513].

Originally a conjecture of Littlewood. The answer is yes (for all $n\geq 2$), proved by Balister, Bollobás, Morris, Sahasrabudhe, and Tiba [BBMST19].

See also [230].

Solved in the affirmative by Barth and Schneider [BaSc72].

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 for all $z\in\mathbb{C}$ 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 $\lvert z\rvert=1$.

See also [228].

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 $c>0$.

This was improved to \[f(n) \leq \exp( cn^{1/3}(\log n)^{4/3})\] by Odlyzko [Od82].

If we denote by $f^*(n)$ the analogous quantity with the assumption that $a_1<\cdots<a_n$ then Bourgain and Chang [BoCh18] prove that \[f^*(n)< \exp(c(n\log n)^{1/2}\log\log n).\]

A reverse Littlewood-Offord problem. Erdős originally asked this with $\sqrt{2}$ replaced by $1$, but Carnielli and Carolino [CaCa11] observed that this is false, choosing $z_1=1$ and $z_k=i$ for $2\leq k\leq n$, where $n$ is even, since then the sum is at least $\sqrt{2}$ always.

Solved in the affirmative by He, Juškevičius, Narayanan, and Spiro [HJNS24]. The bound of $1/n$ is the best possible, as shown by taking $z_k=1$ for $1\leq k\leq n/2$ and $z_k=i$ otherwise.

See also [498].

See also [521] and [522].

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.

This was solved by Schinzel [Sc87], who proved that \[f(k) > \frac{\log\log k}{\log 2}.\] In fact Schinzel proves lower bounds for the corresponding problem with $P(x)^n$ for any integer $n\geq 1$, where the coefficients of the polynomial can be from any field with zero or sufficiently large positive characteristic.

Schinzel and Zannier [ScZa09] have improved this to \[f(k) \gg \log k.\]

A problem of Selfridge and Straus [SeSt58], who prove that this is true if $k=2$ and $\lvert A\rvert \neq 2^l$ (for $l\geq 0$). On the other hand, there are examples with two distinct $A,B$ both of size $2^l$ such that $A_2=B_2$.

More generally, they prove that $A$ is uniquely determined by $A_k$ if $n$ is divisible by a prime greater than $k$. Selfridge and Straus sound more cautious than Erdős, and it may well be that for all $k>2$ there exist $A,B$ of the same size with identical $A_k=B_k$.

(In [Er61] Erdős states this problem incorrectly, replacing sums with products. This product formulation is easily seen to be false, as observed by Steinerberger: consider the case $k=3$ and subsets of the 6th roots of unity corresponding to $\{0,1,2,4\}$ and $\{0,2,3,4\}$ (as subsets of $\mathbb{Z}/6\mathbb{Z}$). The correct problem statement can be found in the paper of Selfridge and Straus that Erdős cites.)

A strong form of the Littlewood-Offord problem. Erdős [Er45] proved this is true if $z_i\in\mathbb{R}$, and for general $z_i\in\mathbb{C}$ proved a weaker upper bound of \[\ll \frac{2^n}{\sqrt{n}}.\] This was solved in the affirmative by Kleitman [Kl65], who also later generalised this to arbitrary Hilbert spaces [Kl70].

See also [395].

Cartan proved this is true with $2$ replaced by $2e$, which was improved to $2.59$ by Pommerenke [Po61]. Pommerenke [Po59] proved that $2$ is achievable if the set is connected (in fact the entire set is covered by a single circle with radius $2$).

Chowla's cosine problem. The best known bound currently, due to Ruzsa [Ru04] (improving on an earlier result of Bourgain [Bo86]), replaces $N^{1/2}$ by \[\exp(O(\sqrt{\log N}).\] The example $A=B-B$, where $B$ is a Sidon set, shows that $N^{1/2}$ would be the best possible here.

Littlewood's conjecture, proved independently by Konyagin [Ko81] and McGehee, Pigno, and Smith [MPS81].

It is trivial that this value is in $[1/2,1)$. Kövári (unpublished) observed that it must be $>1/2$. Clunie and Hayman [ClHa64] showed that it is $\leq 2/\pi-c$ for some absolute constant $c>0$. Some other results on this quantity were established by Gray and Shah [GrSh63].

See also [227].

Boas (unpublished) has proved the first part, that such a path must exist.

Huber [Hu57] proved that for every $\lambda>0$ there is a path $C_\lambda$ such that this integral is finite.

A problem of Pólya. Results of Wiman [Wi14] imply that if $(n_{k+1}-n_k)^2>n_k$ then $\limsup \frac{m(r)}{M(r)}=1$. Erdős and Macintyre [ErMa54] proved this under the assumption that \[\sum_{k\geq 2}\frac{1}{n_{k+1}-n_k}<\infty.\]

A conjecture of Fejér and Pólya. Fejér [Fe08] proved that if $\sum\frac{1}{n_k}<\infty$ then $f(z)$ assumes every value at least once, and Biernacki [Bi28] showed that this holds under the assumption that $n_k/k\to \infty$.

A problem of Turán, who proved that this maximum is $\gg 1/n$. This was solved by Atkinson [At61b], who showed that $c=1/6$ suffices. This has been improved by Biró, first to $c=1/2$ [Bi94], and later to an absolute constant $c>1/2$ [Bi00]. Based on computational evidence it is likely that the optimal value of $c$ is $\approx 0.7$.

Erdős and Offord [EO56] showed that the number of real roots of a random degree $n$ polynomial with $\pm 1$ coefficients is $(\frac{2}{\pi}+o(1))\log n$.

See also [522].

Erdős and Offord [EO56] showed that the number of real roots of a random degree $n$ polynomial with $\pm 1$ coefficients is $(\frac{2}{\pi}+o(1))\log n$.

Solved by Yakir [Ya21], who proved that almost all such polynomials have \[\frac{n}{2}+O(n^{9/10})\] many roots in $\{ z\in \mathbb{C} : \lvert z\rvert \leq 1\}$.

See also [474] and [521].

Salem and Zygmund [SZ54] proved that $\sqrt{n\log n}$ is the right order of magnitude, but not an asymptotic.

A problem of Salem and Zygmund [SZ54]. Chung showed that, for almost all $t$, there exist infinitely many $n$ such that \[M_n(t) \ll \left(\frac{n}{\log\log n}\right)^{1/2}.\] Erdős (unpublished) showed that for almost all $t$ and every $\epsilon>0$ we have $\lim_{n\to \infty}M_n(t)/n^{1/2-\epsilon}=\infty$.

Random polynomials with independently identically distributed coefficients are sometimes called Kac polynomials - this problem considers the case of Radamacher coefficients, i.e. independent uniform $\pm 1$ values. The first problem asks whether $m(f)<1$ almost surely. Littlewood [Li66] conjectured that the stronger $m(f)=o(1)$ holds almost surely.

The answer to both questions is yes: Littlewood's conjecture was solved by Kashin [Ka87], and Konyagin [Ko94] improved this to show that $m(f)\leq n^{-1/2+o(1)}$ almost surely. This is essentially best possible, since Konyagin and Schlag [KoSc99] proved that for any $\epsilon>0$ \[\limsup_{n\to \infty} \mathbb{P}(m(f) \leq \epsilon n^{-1/2})\ll \epsilon.\] Cook and Nguyen [CoNg21] have identified the limiting distribution, proving that for any $\epsilon>0$ \[\lim_{n\to \infty} \mathbb{P}(m(f) > \epsilon n^{-1/2}) = e^{-\epsilon \lambda}\] where $\lambda$ is an explicit constant.

It is unclear to me whether Erdős also intended to assume that $\lvert a_{n+1}\rvert\leq \lvert a_n\rvert$.

It is 'well known' that, for almost all $\epsilon_n=\pm 1$, the series diverges for almost all $\lvert z\rvert=1$ (assuming only $\sum \lvert a_n\rvert^2=\infty$).

Dvoretzky and Erdős [DE59] showed that if $\lvert a_n\rvert >c/\sqrt{n}$ then, for almost all $\epsilon_n=\pm 1$, the series diverges for all $\lvert z\rvert=1$.

Bernstein [Be31] proved that for any choice of $a_i^n$ there exists $x_0\in [-1,1]$ such that \[\limsup_{n\to \infty} \sum_{1\leq i\leq n}\lvert p_{i}^n(x)\rvert=\infty.\] Erdős and Vértesi [ErVe80] proved that for any choice of $a_i^n$ there exists a continuous $f:[-1,1]\to \mathbb{R}$ such that \[\limsup_{n\to \infty} \lvert \mathcal{L}^nf(x)\rvert=\infty\] for almost all $x\in [-1,1]$.