# Quantitative approximate definable choices

## TL;DR

This paper introduces a method to simplify complex geometric selections by allowing approximate choices, reducing computational complexity.

## Contribution

The novel approach allows approximate selections with linear complexity, independent of variable count.

## Key findings

- Approximate selections can be constructed with linear complexity relative to the projection's complexity.
- The method avoids exponential dependence on the number of variables.
- The theory developed has applications to the Sard conjecture in sub-Riemannian geometry.

## Abstract

In semialgebraic geometry, projections play a prominent role. A definable choice is a semialgebraic selection of one point in every fiber of a projection. Definable choices exist by semialgebraic triviality, but their complexity depends exponentially on the number of variables. By allowing the selection to be approximate (in the Hausdorff sense), we improve on this result. In particular, we construct an approximate selection whose degree is linear in the complexity of the projection and does not depend on the number of variables. This work is motivated by infinite–dimensional applications, in particular to the Sard conjecture in sub-Riemannian geometry. To prove these results, we develop a general quantitative theory for Hausdorff approximations in semialgebraic geometry, which has independent interest.