Theoretical analysis reveals lower bounds on distinct distances between points on Pfaffian curves, indicating that algebraic distance bounds extend to transcendental curves.
We generalize Pach and de Zeeuw’s lower bound for distinct distances between points on two curves, from algebraic curves to Pfaffian curves. Pfaffian curves include those that can be defined by any combination of elementary functions, including exponential and logarithmic functions, rational and irrational powers, trigonometric functions and their inverses, integration, and more. The bound remains Ω (min 3/4n3/4,m²,n²\) Ω ( min { m 3 / 4 n 3 / 4 , m 2 , n 2 } ) , as obtained from the proximity technique of Solymosi and Zahl.
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Natarajan et al. (2026) studied this question.
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