Timeline for High sum of fractional parts

9 events

when toggle format	what		by	license	comment
May 16, 2020 at 19:48	comment	added	Ilya Bogdanov		You may replace it with "whenever". But, in fact, in that concrete case it is "iff".
May 16, 2020 at 18:24	comment	added	mathworker21		@IlyaBogdanov what does "as soon as" mean?
May 11, 2020 at 4:37	comment	added	abx		Oh, right of course. Still the curve is dense in some nontrivial subtorus (if at least one of the $x_i$ is irrational). But that makes the proof a bit complicated.
May 10, 2020 at 19:14	comment	added	Ilya Bogdanov		@abx: this is incorrect as soon as $1,x_1,x_2,\dots,x_n$ are linearly dependent over $\mathbb Q$. That's why Kronecker's theorem at my reference looks a bit... involved. In our case, they are dependent, as the $x_i$ sum up to $1$!
May 10, 2020 at 15:05	vote	accept	Dexter
May 10, 2020 at 14:59	comment	added	abx		... another way of saying it is that the curve $t\mapsto (tx_1,\ldots ,tx_n)$ is dense in the torus $\mathbb{R}^n/\mathbb{Z}^n$, thus becomes as close as you want to $0$ for some $t>0$.
May 10, 2020 at 14:50	comment	added	Ilya Bogdanov		I've put a reference, although it is an overkill: one may merely use the pigeonhole principle.
May 10, 2020 at 14:49	history	edited	Ilya Bogdanov	CC BY-SA 4.0	added 100 characters in body
May 10, 2020 at 14:17	history	answered	Ilya Bogdanov	CC BY-SA 4.0