70 problems

Let $A\subseteq\mathbb{R}$ be an infinite set. Say that $A$ is affinely embedded into a set $E\subseteq\mathbb{R}$ if there are real numbers $a\neq 0$ and $b$ such that…

A subset $X\subseteq\mathbb{R}$ is full measure universal if every Lebesgue measurable subset of $\mathbb{R}$ whose complement has Lebesgue measure zero contains an affine copy of…

Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be a $C^1$ map, and let $J(f)$ denote its Jacobian matrix. A point $x$ is a fixed point if $f(x)=x$. The two-fixed-point conjecture. If $f$ has…

Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be a $C^1$ map with $f(0)=0$, and let $J(f)$ denote its Jacobian matrix. The $C^1$ fixed point conjecture. If the eigenvalues of $J(f)$ have abs…

Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be a $C^1$ map, and let $J(f)$ denote its Jacobian matrix. The $C^1$ stability conjecture. If the eigenvalues of $J(f)$ have strictly negative r…

Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be a $C^1$ map, and let $J(f)$ denote its Jacobian matrix. A matrix is unipotent when all its eigenvalues are $1$. The $C^1$ unipotent Jacobian…

Exact value conjecture. The upper bound for $\delta_{\mathcal H}$ obtained in the paper, which is the relevant solution of a cubic equation and is approximately $0.272$, is in fact…

Conjecture. For general $N$, a ratio vector determines unique real numbers $r_{3},\ldots,r_{N}$ such that the polynomial $p$ has that ratio vector.

Yomdin's conjecture. For $k\geqslant n$, a compact set $B=C_vf$ for some function $f\in C^k(\mathbb R^n,\mathbb R)$ with compactly supported derivative if and only if $B$ is a…

Let $R_k$ be the regions introduced in the paper, and let $(a,b)$ denote the parameter pair in Theorem 11.3.2. Sharpness conjecture. The region described in $(a,b)$ by Theorem 11.3…

Let $\vbox{\text to 6.5pt{\kern.5pt\special{psfile=Graph/alpha_ss.ps}\hfill}}$ be the exponent appearing in the definition of the integral functional and let $\mathcal D_{A_1}$ be…

One-dimensional Maxwell bound. For any values of the charges, the function $V_{\alpha}^*(x)$ has at most $2l-1$ real critical points. This is a stronger one-dimensional version of…

Let $\Phi$ be the function considered above, and let $\mathcal{BV}_p(0,2)$ denote the space of functions of bounded $p$-variation on $(0,2)$. The preceding result gives…

Let $\textup{\textsf{open}}$, $\textup{\textsf{HBU}}$, $\mathbb{Z}_{2}^{\omega}$, and $\textup{\textsf{QF-AC}}^{0,1}$ denote the principles and systems used in the paper. Non-impli…

Matkowski's conjecture. The function $f$ solves this equation if and only if there exist $a,b\in\mathbb{R}$ with $a\neq b$ such that

Let $(x_n)$ be a fast convergent sequence, and let $\widehat{E(x_n)}$ denote its adjoint Cantorval. An adjoint Cantorval is achievable if it is the achievement set of some converge…

The $(P_n)$ conjecture. $(P_n)$ is true for all nonnegative integers $n$.

Single-base representation conjecture. For every $a_n,b,c_n,d\in\mathbb{R}$ and $n\in\mathbb{N}$, there exist $u_n,{\Greekmath 010D}\in\mathbb{R}$ such that

A Riemann difference of order $n$ is determined by its nodes and coefficients; it has the Marcinkiewicz-Zygmund (MZ) property when … imply $f(h)=o(h^n)$, where $Rf(h)$ denotes the…

Let $\operatorname{\textnormal{\textinterrobang}}\colon[0,1]\to\mathbb{R}$ be the continuous, strictly increasing extension of the interrobang function. Interrobang differentiabili…

Let $\alpha^{(p)}$ be a quadratic irrational in $\mathbb{Q}_p$ that has an image in $\mathbb{R}$. Consider its continued fraction obtained by Browkin I, Browkin II or Algorithm MR.…

Let $P\subseteq\mathbb{R}$ be a perfect set. Erdős–Kunen–Mauldin strengthening. There exists a set $M\subseteq\mathbb{R}$ with Lebesgue measure zero such that … for all…

A family of subsets of $\mathbb{R}$ is a sigma-ideal if it is closed under taking subsets and under countable unions. Prikry's conjecture. The family of strongly meager subsets of…

For $X\subseteq\mathbb{R}$, call $X$ strongly meager if $X+M\neq\mathbb{R}$ for every Lebesgue measure zero set $M\subseteq\mathbb{R}$. Full measure universality characterization c…

Let $X\subseteq\mathbb{R}$ be topologically universal if every dense $G_\delta$ subset of $\mathbb{R}$ contains an affine copy of $X$. Gallagher–Lai–Weber's conjecture. No uncounta…