Is the sequence sin(n^2), n=1,2,3... dense in [-1,1]?

Question

Does anyone know the answer?

Answer 1

Well, I'll get started on it. But I can't complete the proof yet.

Clearly, the real issue is: "Is the sequence n^2 dense modulo (2*pi)?" If it is, the question posed follows automatically.

I am certain that the answer is "Yes", and I am almost certain that it depends only upon the irrationality of pi (and not on it's transcendental nature).

One approach: Approximate 2*pi as a fraction p/q. For n not too large, n^2 modulo (p/q) will be very close to n^2 module (2*pi) [provided you stay away from 0]. Then the distribution of the sequence is determined by the distribution of q*n^2 modulo (p), among the integers. Can we argue that this will always be well-spaced (not concentrated)? Unfortunately, my memory about the behavior of modulo arithmetic, as a group, is not very strong.

We would need to show that, for n < N(epsilon), there is no gap in coverage as big as epsilon. The smaller epsilon, the bigger the p & q would need to be; but since 2*pi is irrational, there is no limit to the size of the numerator & denominator.

Maybe more steps will occur to me...

Answer 2

o problema é matemático,por que colocá-lo em inglês?
qual o x da questão?
o seno ou o senso?

espero muito ter lhe ajudado.

Answer 3

yes
sinus function has -1<=sinx<=1 values