Formal evidence
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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.
Scope: The checked theorem supplies a positive constant c for which c · q⁻¹³³ᐟ¹⁰ ≤ ‖q log₂ 3‖ for every positive integer q.
Lean checkedBuild passed
Unfinished proof stepsNone
PublicationAccepted formal theorem
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)The theorem at a glance
Rhin phase gap at a glance
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
∃ 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.
Result boundary
What this theorem 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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