Let's take the problem of the backpack: $A_1,\ldots ,A_n$ the weights that are integers, and we want to know if we can achieve a total weight of $V$.

We take $$I=\dfrac{1}{2\pi}\int_0^{2\pi} \exp(-iVt)\times (1+\exp(iA_1t))\times\cdots\times(1+\exp(iA_nt)) \, dt.$$

The question then becomes what $I=0$ or $I\geq 1$.

Can we not approach the value of $I$ with the method of Monte Carlo, or others?

Why are these approaches not succeeding?

Remark : it's not difficult to find a good approximation of $B\times t \bmod 2\pi$, where $B$ is a big integer and $t\in [0,2\pi]$, because we know excellent approximation of $\pi$.