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Let $P$ be any non-linear polynomial with integer coefficients. Consider the function defined by keeping just the Fourier coefficients of the ”fractional part” function whose frequencies are of the shape $P(n)$ with $n$ a natural number.
We will show that the smoothness properties of this function are quite complex in the sense that for each s in some interval, the points with Holder exponent s form a set of positive Hausdorff dimension.
This had been previously shown by S. Jaffard for $P$ a quadratic polynomial, namely for Riemann’s example of a continuous but almost nowhere differentiable function.