The uniformity conjecture

This paper discusses the relationship between the syntactic length of expressions built up from the integers using field operations, radicals and exponentials and logarithms, and the smallness of non zero complex numbers defined by such expressions. The Uniformity Conjecture claims that if the expressions are written in an expanded form in which all the arguments of the exponential function have absolute value bounded by 1, then a small multiple of the syntactic length gives a bound for the number of decimal places needed to distinguish the defined number from zero. The consequences of this conjecture are compared with some known results about closeness of approximation from Liouville, Baker, Waldschmidt, Thue-Siegel-Roth. A few of many practical computational consequences are stated. Also the problem of searching for a possible counterexample to the Uniformity Conjecture is discussed and some preliminary results are given.