MathOverflow

Can one take a parameter to its admissible boundary in a Diophantine argument?
I am teaching myself techniques for Diophantine equations.
Can one take a parameter to the boundary of its admissible region in a Diophantine argument? If not, what does this imply for other proofs that rely on an analogous boundary-limit step?...

A double poset (Malvenuto–Reutenauer, J. Combin. Theory A 118 (2011) 1322–1333) is a finite set $E$ with two partial orders $(E, \le_1, \le_2)$. Foissy (arXiv:1101.5231) uses the involution $\iota(E,\le_1,\le_2) = (E,\le_2,\le_1)$ that exchanges the...

Consider a finite-dimensional Euclidian space $E$, a bounded subset $S \subset E$, with $0 \in S$ and $f: S \rightarrow E$ an $\epsilon$-isometry with $f(0) = 0$. I am interested in a case where $\epsilon$ is a small but sizeable fraction of the diameter...

Is there a version of the following result anywhere in the literature?
Proposition 1. Let $p\in\mathbb{Z}[n]$ satisfy $\deg(p)\geq2$, and let $\varepsilon>0$. Then there is $N\in\mathbb{N}$ such that the set $\{p(n);n\in\mathbb{Z}\}$ intersects at most $\varepsilon N$ residue classes mod $N$....