MathOverflow

The theory of locally presentable categories, accessible categories and accessible functors is a powerful theory that includes many large categories and functors of interest. This theory often gives useful explicit descriptions for objects of these categories...

Let $K$ be a compact subset of $\mathbb{R}^2$ with positive (2-dimensional) Lebesgue measure. Is it true that $K$ contains a bi-Lipschitz copy of a set of the form $A\times B$, where $A$ and $B$ are positive (1-dimensional) Lebesgue measure subsets of...

Let $A\in M_n(\mathbb{C})$ be such that$$W(A)=\{\langle Ax,x\rangle : x\in\mathbb{C}^n,\ \|x\|=1\}\subset\Omega,$$where $\Omega\subset\mathbb{C}$ is a convex domain. Let $f$ be a holomorphic function on a neighborhood of $\Omega$. If$$\Re f(z)\ge 0,...

For a prime $p$, define $[a, b](x) = \frac{a - b}{2}x^{(p + 1)/2} + \frac{a + b}{2} x$. That is, $[a, b](x) = ax$ if $x$ is a quadratic residue and $[a, b](x) = bx$ is $x$ is a quadratic nonresidue. These quadratic maps are non-affine if $a \neq b$.
I want to show that there are infinitely many primes $p$ for which there exists a permutation of $\mathbb{F}_p$ given by non-affine quadratic map $[a, b](x)$ with the following three properties:...