On a rigidity property for quadratic Gauss sums

Let $N$ be a large prime and let $c>1\mathrm{/}4$. We prove that if $f$ is a $\pm 1$-valued completely multiplicative function, such that the exponential sums $$S_f(a):=\sum _{1\le n<N}f(n)e(na\mathrm{/}N),a(\mathrm{m}\mathrm{o}\mathrm{d}N)$$ satisfy the ``Gauss sum-like’’ approximate dilation symmetry property $$\frac{1}{N}\sum _{a(\mathrm{m}\mathrm{o}\mathrm{d}N)}\mathrm{\mid }{S}_f(ap)-f(p)S_f(a){\mathrm{\mid }}^2=o(N),$$ uniformly over all primes $p\le N^c$ then $f$ coincides with a real character modulo $N$ at all but $o(N)$ integers $1\le n<N$. 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.