Modified Weyl–Berry conjecture for Minkowski measurable fractal strings

Let $\mathcal{L}$ be a fractal string with Minkowski dimension $D\in(0,1)$, and let $N_\nu(x)$ be its eigenvalue counting function. Write

$$W(x)=|\Omega|_1x$$

for the Weyl term. If $\mathcal{L}$ is Minkowski measurable, denote its Minkowski content by $\mathcal{M}$.

Modified Weyl–Berry conjecture. There is a positive constant $c_D$ depending only on $D$ such that

$$N_\nu(x)=W(x)-c_D\mathcal{M}x^D+o(x^D),\qquad x\to+\infty.$$

The conjecture asks when the Weyl asymptotic formula has a genuine monotonic second term proportional to $x^D$. The corresponding estimate is known under suitable Minkowski-content hypotheses, but the asserted asymptotic formula for every Minkowski measurable fractal string is presented here as a conjecture.