The integer square root is negligible relative to its input

Proved

ThreeSum.Scales.natSqrt_ratio_tendsto_zero

by wurtle · Oct 14, 2026 · Mathlib 0df444a (Lean v4.33.1)

3sumalgorithmsfine-grained-complexitythree-sum-1962word-ram

Let $\lfloor\sqrt m\rfloor$ denote the natural-number square root of $m\in\mathbb N$ . After coercing numerator and denominator to the reals,

\lim_{m\to\infty}\frac{\lfloor\sqrt m\rfloor}{m}=0.

This limit controls the lower-order square-root rounding term in the construction's scale parameters.

Preamble

import Definitions.Def_ThreeSum1962Source_Bundle07
import Lean.Elab.Tactic.Omega
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Algebra.Order.Floor.Div
import Mathlib.Analysis.Complex.ExponentialBounds
import Mathlib.Analysis.Convex.Deriv
import Mathlib.Analysis.SpecialFunctions.BinaryEntropy
import Mathlib.Analysis.SpecialFunctions.Log.Base
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Nat.Choose.Central
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.Data.Nat.Sqrt
import Mathlib.Data.Real.Basic
import Mathlib.NumberTheory.Bertrand
import Mathlib.NumberTheory.Chebyshev
import Mathlib.Order.Monotone.Basic
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Positivity
import Mathlib.Tactic.Ring


set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option Elab.async false

Formal statement

theorem ThreeSum.Scales.natSqrt_ratio_tendsto_zero : @Filter.Tendsto.{0, 0} Nat Real
  (fun (m : Nat) =>
    @HDiv.hDiv.{0, 0, 0} Real Real Real (@instHDiv.{0} Real (@DivInvMonoid.toDiv.{0} Real Real.instDivInvMonoid))
      (@Nat.cast.{0} Real Real.instNatCast m.sqrt) (@Nat.cast.{0} Real Real.instNatCast m))
  (@Filter.atTop.{0} Nat Nat.instPreorder)
  (@nhds.{0} Real
    (@UniformSpace.toTopologicalSpace.{0} Real (@PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))
    (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero))) := by
  sorry

Source

https://github.com/c0ldzy17/three-sum-lean/blob/2b639655e07421b9ac743de9090459bab473a7c0/Submission/ThreeSum/Scales.lean#L339-L353

View graph