Uniformity Conjecture

The uniformity conjecture postulates a relationship between the syntactic length of expressions built up from the natural numbers using field operations, exponentials, and logarithms, and the smallest of nonzero 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 (Richardson 2002). Richardson (2002) has systematically searched for counterexamples, but not found any.

REFERENCES:

Richardson, D. "The Uniformity Conjecture." In Computability and Complexity in Analysis (Swansea, 2000). Berlin: Springer-Verlag, pp. 253-272, 2001.

Richardson, D. "Testing the Uniformity Conjecture." Draft, 2002. https://www.bath.ac.uk/~masdr/testu.ps.

van der Hoeven, J. "Automatic Numerical Expansions." In Proc. Conference 'Real Numbers and Computers,' Saint-Étienne, France (Ed. J.-C. Bajard, D. Michleucci, J.-M. Moreau, and J.-M. Muller). pp. 261-274, 1995.

van der Hoeven, J. Automatic Asymptotics. Ph. D. thesis. Ecole Polytechnique, 1997.

van der Hoeven, J. "Simultaneous Approximation of Numbers Connected with the Exponential Function." J. Austral. Math. Soc. 25, 466-478, 1978.