Logarithmic--exponential preparation in sharply o-minimal structures
Abstract
We develop a complex-analytic approach to the model theory of the (real) unrestricted exponential. As a consequence we derive sharp forms of many of the foundational results for the structure ℝREexp (and more general structures). In particular we establish sharp o-minimality, a sharp form of Wilkie's theorem of the complement, a sharp form of Wilkie's conjecture and a sharp form of piecewise definability by terms. Our approach is based on a complexification of the LE-preparation theorem of Lion--Rolin. We also develop a parallel complex theory for ℝan,exp, proving for example that the rational points of height H on a nowhere-dense definable set can be interpolated by an algebraic hypersurface of degree poly(log H). This generalizes a theorem of Cluckers--Pila--Wilkie who proved the same statement for ℝanpow.
Related
- Logarithmic exponential sums with multiplicative coefficients
- Logarithmic coefficients for exponential classes of starlike and convex functions
- Generalized Logarithmic Sobolev inequalities on Bounded Domains and exponential decay to Boltzmann states for Nernst--Planck--Navier--Stokes systems
- Logarithmic Sobolev and exponential inequalities for submanifolds in Cartan--Hadamard manifolds
- Michell-Prager type truss structures constructed from integrable discrete power function and discrete logarithmic function
BibTeX
@misc{binyamini2026logarithmicexponential,
title = {{Logarithmic--exponential preparation in sharply o-minimal structures}},
author = {Gal Binyamini and Oded Carmon and Dmitry Novikov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20668},
eprint = {2609.20668},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20668},
}
RIS
TY - UNPB
TI - Logarithmic--exponential preparation in sharply o-minimal structures
AU - Gal Binyamini
AU - Oded Carmon
AU - Dmitry Novikov
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20668
UR - https://arxiv.org/abs/2609.20668
AB - We develop a complex-analytic approach to the model theory of the (real) unrestricted exponential. As a consequence we derive sharp forms of many of the foundational results for the structure ℝ^RE_exp (and more general structures). In particular we establish sharp o-minimality, a sharp form of Wilkie's theorem of the complement, a sharp form of Wilkie's conjecture and a sharp form of piecewise definability by terms. Our approach is based on a complexification of the LE-preparation theorem of Lion--Rolin. We also develop a parallel complex theory for ℝ_an,exp, proving for example that the rational points of height H on a nowhere-dense definable set can be interpolated by an algebraic hypersurface of degree poly(log H). This generalizes a theorem of Cluckers--Pila--Wilkie who proved the same statement for ℝ_an^pow.
ER -