While it is well known that the sawtooth function has a point-wise convergent
Fourier series, the rate of convergence is not the
best possible for the application of approximating the mod function
in small intervals around multiples of the modulus.
We show a different sine series, such that the
sine series of order n has error O(epsilon^(2n+1)) for approximating
the mod function in epsilon-sized intervals around multiples of the modulus.
Moreover, the resulting polynomial, after Taylor series approximation of the
sine series, has small coefficients, and the whole polynomial can be computed
at a precision that is only slightly larger than
-(2n+1)log epsilon, the precision of approximation being sought. This polynomial can then be used to approximate the mod function to almost arbitrary precision,
and hence allows practical CKKS-HE bootstrapping with arbitrary precision.

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