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Diophantine approximation asks how closely a real or complex quantity can be approximated by rational numbers or algebraic numbers of bounded arithmetic complexity. For a real $\alpha$, one compares $|\alpha -p/q|$ with functions of the denominator $q$.

For example, continued fraction convergents to $\sqrt{2}$ give approximations $3/2,7/5,17/12,\dots$. A nonzero integer norm of $p-q\sqrt{2}$ also supplies a lower bound, preventing arbitrarily exceptional accuracy. The Liouville approximation theorem extends this idea to algebraic numbers, while Roth's theorem sharpens the exponent for algebraic irrational inputs. Approximation quality can therefore distinguish algebraic from transcendental behavior, not merely give convenient decimal values.

Michel Waldschmidt works on transcendental number theory and Diophantine approximation. With Patrice Philippon, he wrote quantitative results for linear forms in logarithms of algebraic numbers of algebraic numbers.^[ref]

A nonzero expression such as $b_1\log {\alpha }_1+b_2\log {\alpha }_2$ can be small, but lower bounds limit how small it can be in terms of the algebraic data and coefficient sizes. Such bounds convert qualitative nonvanishing into numerical restrictions useful in solving Diophantine equations. The algebraic-coefficient and integer-coefficient cases require distinct estimates.^[ref]

Patrice Philippon works on transcendental number theory. His joint chapter with Michel Waldschmidt gives lower bounds for linear forms in logarithms of algebraic numbers, explicitly distinguishing general algebraic coefficients from rational-integer coefficients.^[ref]

The central arithmetic question is quantitative: once a logarithmic combination is known to be nonzero, how closely can it approach zero? An explicit bound links Diophantine approximation to finite calculations and can help restrict candidate solutions of Diophantine equations.