On a rigidity property for quadratic Gauss sums

Let $N$ be a large prime and let $c > 1/4$. We prove that if $f$ is a $\pm 1$-valued completely multiplicative function, such that the exponential sums $$S_f(a) := \sum_{1 \leq n < N} f(n) e(na/N), \quad a \pmod{N}$$ satisfy the ``Gauss sum-like'' approximate dilation symmetry property $$\frac{1}{N}\sum_{a \pmod{N}} |S_f(ap) - f(p)S_f(a)|^2 = o(N),$$ uniformly over all primes $p \leq N^c$ then $f$ coincides with a real character modulo $N$ at all but $o(N)$ integers $1 \leq n < N$. As a consequence, taking $f$ to be the Liouville function we connect this exponential sums property to the location of real zeros of $L(s,\chi)$ close to $s = 1$, for $\chi$ the Legendre symbol modulo $N$. Assuming the $L$-functions of primitive Dirichlet characters modulo $N$ have a sufficiently wide zero-free region (of Littlewood type), we also show a more general result in which any $c > 0$ may be taken.