Height Rigidity for Entire Functions

Abstract

We prove that algebraic values of bounded degree and polynomially bounded height are sparse on rational translates of the graph of a transcendental entire function. More precisely, for fixed $\theta \in \overline{\mathbb{Q}}\cap \mathbb{R}$, $D\ge 1$, $t>0$, and for every $\epsilon >0$, only $O(Q^{\epsilon })$ rationals $r\in [0,1]$ of height at most $Q$ can satisfy simultaneously $[\mathbb{Q}(f(\theta +r)):\mathbb{Q}]\le D$ and $H(f(\theta +r))\ll H(r{)}^t$. Consequently, if these bounds hold for every rational $r$ of sufficiently large height, then $f\in \overline{\mathbb{Q}}[z]$ and $°f\le t$. As applications, we obtain rigidity results for entire functions taking rational or bounded-degree algebraic values with polynomially controlled arithmetic height. In particular, this excludes the polynomial-denominator scenario that arises naturally in connection with Mahler's problem on Liouville numbers. The proof combines Pila's bounded-degree counting theorem with standard height estimates and a simple geometric analysis of transcendental entire graphs.