A Disk-Growth Remez Principle and a Modular Proof of the Measurable Tur\'an-Nazarov Inequality

We give a modular proof of the measurable Tur\'an-Nazarov inequality for exponential polynomials. The proof first establishes a Remez principle for holomorphic functions satisfying two disk-growth assumptions. The global growth assumption controls the number of relevant zeros, while the local growth assumption gives an effective degree. This yields Cartan coverings, sublevel estimates, and a geometric-mean Remez inequality. For exponential polynomials with bounded spectral diameter, the required disk growth follows from the classical interval Tur\'an inequality. For large spectral diameter, we use a first-order pruning step. If $\rho = \diam(\spec p)$ and $a\in\spec p$, then $$ Q_a = \rho^{-1}(D-a)p $$ has one fewer exponential term, and the quotient $Q_a/p$ satisfies an absolute weak distribution estimate away from the zero set of $p$. Writing $$ Q_a = \rho^{-1}(D-a)p, \quad Q_b = \rho^{-1}(D-b)p $$ for two farthest spectral points $a,b$ gives $$ Q_a-Q_b = \frac{b-a}{\rho}p, \quad |b-a| = \rho, $$ and hence $|p|\le |Q_a|+|Q_b|$. The induction is carried out in geometric-mean form on the original measurable set. This avoids losing a fixed proportion of the set at each step and gives the classical measurable Tur\'an-Nazarov inequality with the sharp algebraic exponent $m-1$. The final measurable $L^\infty$ estimate is classical; the point here is the modular proof and the geometric-mean induction. The only Tur\'an-type input is the classical interval Tur\'an inequality.