The 36π/127 conjecture
Update: the conjecture is apparently wrong.
There is a problem that bothers me for a long time. Consider function f(x), defined in the following way:
$$f(x) = \sum_{n=0}^\infty \frac{sin(2^n x)}{2^n}.$$
The special thing about f(x) is that it has no derivatives. It may be easily seen from it's graph:
Red graph denotes f(x), gray lines are sin(2nx)/2n.
There is a problem that bothers me for a long time. Consider function f(x), defined in the following way:
$$f(x) = \sum_{n=0}^\infty \frac{sin(2^n x)}{2^n}.$$
Obviously, this sum converges for every real x; and for every x, -2 < f(x) < 2; and f(x) is continuous (due to the uniform convergence of partial sums).
Red graph denotes f(x), gray lines are sin(2nx)/2n.
The red line in the graph is f(x), is obviously a fractal. Using box-counting approach, it's dimension may be approximated as 1.03, though this value tends to decrease, when minimal box size is decreased.
Using numeric methods, I have found that f(x) achieves it's maximal value at point xmax = 0.8905302... (dash line at the graph above) which effectively equals to \(x_0=\frac{36}{127}\pi\). I am looking for the proof of this fact, but without success.
Following plot may give a key to the solution. It is a logarithmic graph, showing how the function behaves in the close vicinity of the point \(x_0\).
The notable feature of this graph is its periodicity. At other points, f(x) have more complex behavior:
Periodical structure appears only if x is a rational fraction of \(\pi\). For other values, graph shows chaotic structure:
Following plot may give a key to the solution. It is a logarithmic graph, showing how the function behaves in the close vicinity of the point \(x_0\).
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| Graph of the f(x) in the vicinity of the point x0. |
Periodical structure appears only if x is a rational fraction of \(\pi\). For other values, graph shows chaotic structure:
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