Given an entire function $f$ of finite order $\unicode[STIX]{x1D70C}$ and positive lower order  $\unicode[STIX]{x1D706}$ , Boxall and Jones proved a bound of the form $C(\log H)^{\unicode[STIX]{x1D702}(\unicode[STIX]{x1D706},\unicode[STIX]{x1D70C})}$ for the density of algebraic points of bounded degree and height at most $H$ on the restrictions to compact sets of the graph of  $f$ . The constant $C$ and exponent $\unicode[STIX]{x1D702}$ are effectively computable from certain data associated with the function. In this followup note, using different measures of the growth of entire functions, we obtain similar bounds for other classes of functions to which the original theorem does not apply.

给定整个函数 $ f $ 的有限阶 $ \ unicode [STIX] {x1D70C} $ 和低阶正数  $ \ unicode [STIX] {x1D706} $ ，Boxall和Jones证明了形式 为$ C（\ log H ）^ {\的unicode [STIX] {x1D702}（\的unicode [STIX] {x1D706}，\的unicode [STIX] {x1D70C}）} $ 有界度的代数点的密度和高度至多 $ H $ 上对 $ f $ 图的紧集的限制 。常数 $ C $ 和指数 $ \ unicode [STIX] {x1D702} $ 可以从与该功能关联的某些数据中有效地计算得出。在此后续笔记中，使用对整个函数增长的不同度量，我们获得了不适用于原始定理的其他类函数的相似范围。