Diophantine approximation · formal theorem

Power-Law Phase Gap for Multiples of log₂ 3

There exists a positive constant c such that every positive integer q satisfies c · q⁻¹³³ᐟ¹⁰ ≤ ‖q log₂ 3‖, the distance to the nearest integer.

Lean proofpassed

Unfinished proof stepsNone

Formal resultAccepted

Exact scope: The checked theorem supplies a positive constant c for which c · q⁻¹³³ᐟ¹⁰ ≤ ‖q log₂ 3‖ for every positive integer q.

Line counts exclude blank lines; comments and documentation count. The total is the commit-pinned first-party Lean import closure; Mathlib and other third-party dependencies are excluded.

The theorem at a glance

Rhin phase gap at a glance

Ivory editorial poster for the Rhin phase-gap theorem, with the global nearest-integer inequality, one identified integer phase and scattered fractional phases around a modulo-one circle, large-height and finite-height proof movements, and exact scope. — Rhin's large-height estimate and finite absorption produce one positive global power-law phase-gap witness.

Accessible transcript

∃ c > 0, ∀ q ≥ 1, c · q^(-133/10) ≤ ‖q log₂ 3‖

The proof controls all sufficiently large q through an explicit linear-form estimate, takes a positive normalized minimum over the finitely many smaller q, and chooses one witness c valid in both ranges.

Read the complete poster transcript

DIOPHANTINE APPROXIMATION · LOG₂ 3

A POWER-LAW GAP FROM THE INTEGERS

THERE EXISTS c > 0 SUCH THAT FOR EVERY q ≥ 1

c · q^(-133/10) ≤ ‖q log₂ 3‖

‖x‖ = distance from x to the nearest integer

0 ≡ 1 mod 1

× log₂ 3 mod 1

≥ c · q^(-133/10)

HOW THE PROOF MOVES

1 · LARGE HEIGHTS

Rhin's explicit linear-form estimate controls every sufficiently large q.

2 · FINITE HEIGHTS

The finitely many smaller positive q have a positive minimum normalized gap.

3 · ABSORB THE TWO RANGES

Take one positive witness c that works globally.

EXACT SCOPE

c is existential, not numerically exposed. The gap shrinks with q. This does not assert a fixed separation, equidistribution, or a Collatz theorem. The downstream exponent condition is strict: d < 5/143.

Theorem schematic

A global power-law phase gap

Cobalt irrational-rotation samples remain outside gold neighborhoods of an aligned integer lattice; the protected widths shrink by a power law while finite and large-height regions join into one global band. — One positive witness keeps every positive multiple of log₂ 3 at least a power-law distance from the nearest integer.

∃ c > 0, ∀ q ≥ 1, c · q^(-133/10) ≤ ‖q log₂ 3‖

There exists c > 0 such that c · q^(-133/10) is no greater than the nearest-integer distance of q log₂ 3 for every q ≥ 1. The witness is existential and the separation shrinks with q.

Exact formal proposition

Hypotheses and conclusion

This is the exact recorded theorem statement, followed by the source definitions that give its named predicates and quantities their full meaning.

theorem existsPhaseGapRhin : ∃ c : ℝ, PhaseGap c (143 / 10 : ℝ)

Definitions used in this proposition

`PhaseGap`

structure PhaseGap (c kappa : ℝ) : Prop where
  c_pos : 0 < c
  c_le_one : c ≤ 1
  two_lt_kappa : 2 < kappa
  gap : ∀ q : ℕ, 0 < q →
    c * Real.rpow (q : ℝ) (1 - kappa) ≤
      nearestIntegerNorm ((q : ℝ) * logTwoThree)

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Result boundary

What this page does—and does not—establish

The checked theorem supplies a positive constant c for which c · q⁻¹³³ᐟ¹⁰ ≤ ‖q log₂ 3‖ for every positive integer q.

The witness c is existential; the theorem exposes no numerical value for it.
The power-law gap shrinks with q, so this is not a fixed uniform separation from the integers.
This supporting theorem does not itself prove a Collatz or density result.
Downstream natural-density rates use a strict exponent cap d < 5/143; the cap is not attained.

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Formal-result publication and review details

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