An argument for error accumulation during complex DFT
Since the fourier coefficients might in general be complex numbers $xi + y$ with $x$ or $y$ or both irrational, a standard finite precision computer of course will not output the exact fourier coefficients. However, if you want to claim that you cannot get a good approximation, you are wrong.

Look at this answer. Markus shows that if you want to compute DFT of a vector of length $n$ up to $b$ bits of precision, then you need floating point numbers with $m$ bits where $m = \log n + \log \log n + b + C$ for $C$ some fixed constant. This is a fairly good bound. Also, there is no reason why the double type of the C language should be any hard limitation if you care about precision: there are libraries for high-precision arithmethic, and it's not even all that hard to code your own procedures to multiply and add floating point numbers of fixed high (higher than that of double) precision. Markus's answer shows that you won't lose much in running time: the running time will be roughly $n \log n (b/\log n)$ (for $b &gt; \log n$) arithmetic operations on standard C types.