TeXipedia

lesssim

Denotes an inequality that is approximately less than, combining the concepts of 'less than' and 'approximately equal to' into a single symbol.

Overview

Commonly used in mathematical analysis, numerical computations, and theoretical proofs where precise bounds are not required or available.

  • Particularly useful in asymptotic analysis and complexity theory to express upper bounds.
  • Appears frequently in estimation problems and approximation theory.
  • Often employed when discussing algorithmic efficiency or error bounds.
  • Serves as a more precise alternative to informal notation like 'roughly less than' in technical writing.

Examples

Comparing growth rates in computational complexity analysis.

T(n)≲nlog⁡nT(n) \lesssim n\log n 
T(n) \lesssim n\log n

Expressing an approximate inequality in statistical bounds.

P(X>x)≲exp⁡(−x2/2)P(X > x) \lesssim \exp(-x^2/2) 
P(X > x) \lesssim \exp(-x^2/2)

Describing asymptotic behavior in mathematical analysis.

f(x)≲x2 as x→∞f(x) \lesssim x^2 \text{ as } x \to \infty 
f(x) \lesssim x^2 \text{ as } x \to \infty