
analysis question on periodic functions
Posted:
Aug 17, 2013 6:21 AM


Suppose we have an odd continuous function f: R > R with period 1 so that f(x+1) = f(x), and f(x) = f(x).
Suppose f has mean zero over the unit interval, so int_{0, 1} f(x) dx = 0, but that int_{0, 1}  f(x)  dx > 0 (so it's not constantly zero).
Given a real number a> 0, consider the series:
f(a) + f(2a) + f(3a) + ...
with partial sums
S_k (a) = sum_{j = 1 ... k} f(ka).
===
(i) If a is rational, is it necessarily true that the S_k (a) are bounded in absolute value?
if the answer to (i) were YES, then there's also (ii) below: if the answer were NO, (ii) might be forgotten.
(ii) If the partial sums S_k (a) are bounded for some real number a>0, then does it follow that a is a rational number?
 abc?

