The growth of elementary functions (Q532079)

Let \(f\) be a function defined by an expression involving real constants, the functions \(x, e^x, \log x\), algebraic operations and compositions. By a classical theorem of Du Bois-Reymond, such a function cannot grow faster than an iterated exponential. Surprisingly, the author together with Ruzsa showed this is no longer the case if the class of functions is widened to include the sine function and its inverse [\textit{M. Laczkovich} and \textit{I. Z. Ruzsa}, Ill. J. Math. 44, No.~1, 161--182 (2000; Zbl 0948.26002)]. The present paper limits this by proving that such a function must be dominated by an iterated exponential except on a set of finite measure. As is often the case, this result is obtained by proving a more general result that needs some setting up. Let \(\mathcal A\) be the class of real analytic functions defined on the open intervals of \(\mathbb R\) and let \(IA\), \(IIA\) be the classes functions in \(\mathcal A\) that are respectively integrals of algebraic functions, functions with inverses that are integrals of algebraic functions; put \(B=IA\cup IIA\). \(IA\) includes the constant functions, \(x,\log x,\arcsin x\) (defined on \((-1, 1)\)), and \(IIA\) includes \(\sin x, e^x\). The extended class of elementary functions is the smallest class of functions, \(E^*\), containing \(B\) and such that (i) if \(f\in \mathcal A\) and there are functions \(g,h\in E^*\) such that \(f= g\circ h\big| _{\operatorname{dom}f}\), then \(f\in E^*\); (ii) \(E^*\) is closed under differentiation; (iii) if \(f,g\in E^*\) then \(f-g\), \(fg\), \(f/g\) restricted to any interval in \(\operatorname{dom}f\cap\operatorname{dom}g\) is in \(E^*\); (iv) if \(g\) is such that \(f_ng^n +\dots+ f_1g + f_0 = 0\) for some \(f_0,\dots, f_n\in E^*\), then \(g\in E^*\). It is for this extended class of elementary functions that the main result mentioned above is obtained after what is a very intricate argument. Reference must be made to the paper for this argument as well as for a more careful description of the extended class of elementary functions.