OK -- let me somewhat amend my previous post -- though the conclusions still seem valid to me:
I showed: the average value for
sum (sin(c+n)^n) is about sqrt(N/pi) n=0..N
but of course, that by itself doesn't even preclude the sum converging to 0 almost everywhere (that's a standard elementary real analysis question.)Still, it's at least suggestive!
Suggestive too, is that sin(theta) > (1/2)^(1/n) with measure about C/sqrt(n). [initially I estimated by using just 1 - (1/2)^(1/n); that give only ln(n)/n) --- which is still sufficient --- but taking the arcsin for values up near 1 improves that considerably!]
This *really* strongly suggests that for almost all points c there will be an infinite number of n for which sin(c+n)^(1/n) is bigger than 1/2. It still doesn't *prove* it, of course, but it's almost there -- and when you consider that the integer translations of pi/2 on the unit circle form a dense set it seems even more suggestive...