What is the Convergence of |Sin(n)|?

[Bacle]

Hi, All:

I am trying to see if a_n:={|Sin(n)|}, with n=1,2,... and | . | standard absolute value,
is convergent. I know the set {k.pi}, k=1,2,... is dense in [0,1] (pi is equidistributed mod1) , and we have that Sin(n)=Sin(n+pi), but it seems like {|Sinn|} is dense in [0,1], so that it cannot have a limit (i.e., a unique limit point). Any Ideas?

 

[Josh Swanson]

No, it doesn't converge. From the equidistribution theorem, a result very similar to the one you quoted holds: {k/(2pi) mod 1} is dense in [0, 1]. Scaling everything up by 2pi, {k mod 2pi} is dense in [0, 2pi]. Now 0 and pi/2 are both limit points of this set, so they're approached by convergent sequences s1 and s2 respectively. But then the limit of |sin(an)| on these two sequences is |sin(0)|=0 and |sin(pi/2)|=1 respectively, from continuity. The lim sup and lim inf are then unequal, so the original sequence doesn't converge. This same reasoning does show that {|sin(n)|} is dense on [0, 1].

 

[Bacle]

Thanks, Josh; a new question came to me:

Is there a way of determining any actual value assumed by {|sin(n)|}? Or, are we sure there are rationals or irrationals in the sequence? We can of course exclude {|sinx|: x irrational, and values like Sqr2/2 , as the image of Pi/4; do you know of any result, e.g.,
like with elements of the Cantor set and the base-3 representation (excluding, I think, strings with a 0 in them) about this set?

 

[Josh Swanson]

The only relevant result I'm aware of is the http://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem" (actually, Lindemann's half is sufficient) which says that e^a where a != 0 is any algebraic number is transcendental. [Briefly, an algebraic number is a root of some polynomial equation with integer coefficients. A transcendental number is not. Transcendental numbers are in particular irrational, since the root of ax - b is b/a.] It turns out that any rational polynomial with integer coefficients evaluated at a transcendental number gives a transcendental result. I don't know a name for this result. The only proof I've seen is my own, since it was an exercise in a Galois theory book of mine.

In any case, if n is a positive integer, ni is algebraic [(x - ni)(x + ni) = x^2 + n^2], so e^(ni) is transcendental, so

(e^(in))^2 - 1)/(e^(in))
= e^(in) - e^(-in)

is transcendental, so

1/(2i) (e^(in) - e^(-in))
= sin(n)

is transcendental, so |sin(n)| is transcendental. This immediately rules out eg. sqrt(2)/2, since this is obviously algebraic. An arbitrary finite nesting of radicals is also ruled out: sqrt(2 + sqrt(2))/2 is never hit, for instance.


To be clear the sequence {|sin(n)|} where n is a positive integer is composed only of transcendental numbers, so contains no irrationals. You can actually use the formula listed http://en.wikipedia.org/wiki/List_o....2C_cosine.2C_and_tangent_of_multiple_angles" to write sin(n) in terms of sin(1), cos(1) = sqrt(1 - sin^2(1)), and positive exponents of these two. That is, sin(n) is in the field extension Q(sin(1), cos(1)).

 

[Bacle]

Excellent; very helpful, Josh. Thanks.