Timeline for Counting primitive solutions to a diophantine inequality

8 events

when toggle format	what		by	license	comment
Dec 28, 2018 at 4:07	vote	accept	Itay
Dec 27, 2018 at 1:54	answer	added	Rodrigo		timeline score: 0
Dec 21, 2018 at 23:02	comment	added	Andreas Rüdinger		I think I have an heuristic argument that for $\tau =(\sqrt{5}-1)/2$ the number of solutions scales as $S(\tau,N) \sim N^{1-\alpha}$. My argument makes use of the fact that you can expand a natural number $q$ in a Fibonacci base ($ q = \sum c_k f_k$, where $f_k$ are Fibonacci numbers and $c_k \in \{0,1\}$ with $c_k c_{k+1}=0$), you then can get estimates for the difference of $\tau q$ to the next integer rather easily. Unfortunately I don’t have the time to write it up properly. And the argument seems to work only for quadratic irrationalities. $\alpha \to 1$ gives correctly $\sim \log N$.
Dec 21, 2018 at 17:24	history	edited	Itay	CC BY-SA 4.0	added 215 characters in body
Dec 21, 2018 at 15:12	history	edited	Itay	CC BY-SA 4.0	edited body
Dec 21, 2018 at 7:10	history	edited	Itay	CC BY-SA 4.0	deleted 3 characters in body
Dec 21, 2018 at 6:50	history	edited	Itay	CC BY-SA 4.0	edited title
Dec 21, 2018 at 6:43	history	asked	Itay	CC BY-SA 4.0