Derivatives of the discrete maximal function of a characteristic function beat the constant 1/2 from order eight on
Abstract
For a set S ⊆ ℤ let Mᵘ_(ℤ)χ_S be its uncentered discrete Hardy–Littlewood maximal function and let f^((k)) denote the k-th forward difference. Liao, Madrid, Palsson and Weigt define C_(k,p) as the least constant with norm(Mᵘ_(ℤ)χ_S)^((k))_(lp p(ℤ)) ≤ C_(k,p)normχ_S^((k))_(lp p(ℤ)) for every S, determine it for k ≤ 2, and for k ≥ 3 record only 1/2 ≤ C_(k,p) together with an exponential lower bound whose value first exceeds 1/2 at k = 36 (p ∈ {1,∞}) and at k = 23 (p = 2); their experiments could not improve on 1/2 for 3 ≤ k ≤ 6. We show that the constant 1/2 is beaten from order eight on: the three-point set {0,1,4} on the circle ℤ₇ has lp1 ratio exactly 13/25 at k = 8, and the four-point set {0,1,4,7} on ℤ₁₀ has lp1 ratio 97/186 and squared lp2 ratio 23083/90504 > 1/4 at k = 8 and lp∞ ratio 139/270 at k = 9. Through the circle-to-line lemma of the same authors this gives C_(8,1) > 1/2, C_(8,2) > 1/2 and C_(9,∞) > 1/2. The seven-point set alone keeps its ratio above 1/2 at every order from 8 to 35 in lp1, from 9 to 22 in lp2 and from 11 to 35 in lp∞, which is exactly where the exponential bound takes over; hence C_(k,1) > 1/2 and C_(k,2) > 1/2 for every k ≥ 8 and C_(k,∞) > 1/2 for every k ≥ 9, with no order left over. Below order eight the wall stands: for every subset of every circle of length at most 15 and every 2 ≤ k ≤ 7 the lp1 and lp∞ ratios are at most 1/2, one order beyond the range the earlier experiments report. Every ratio is an exact rational and every statement is machine-checked in Lean 4; the question whether C_(k,p) > 1/2 for 3 ≤ k ≤ 6 remains open.
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Source snapshot 2026-09-07 03:53 UTC
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Claim ledger
Stated results
MC1known2026-09-03
M_(Z₃) chi₀ = (1, 1/2, 1/2), so the circle ratio is exactly 1/2 at every k >= 1 – the source's thm:ckp witness
MC2routine2026-09-03
The two implementations of M_(Z_N) – the literal supremum over s+t+1 <= 2N and the incremental form the sweeps run – agree on all 1023 subsets of circles of length <= 9
MC3known2026-09-03
The period-3 set 0,3,6,9,12 in Z₁5 attains the ratio exactly 1/2 at every k <= 8 in l¹ and lⁱnf; the sweeps are non-vacuous
MW1routine2026-09-03
M_(Z₇) chi₀,1,4 = (1, 1, 2/3, 3/5, 1, 3/5, 2/3), exactly
MW2routine2026-09-03
M_(Z₁0) chi₀,1,4,7 = (1, 1, 2/3, 3/5, 1, 1/2, 1/2, 1, 3/5, 2/3), exactly
MW3candidate2026-09-03
||(M_(Z₇) chi₀,1,4)^((8))||₁ / ||chi^((8))||₁ = 13/25 > 1/2, hence C_(8,1) > 1/2 by the source's Lemma lem:discretevsperiodic
MW4candidate2026-09-03
||(M_(Z₁0) chi₀,1,4,7)^((8))||₁/||chi^((8))||₁ = 97/186 and (||.||₂/||.||₂)² = 23083/90504 > 1/4, hence C_(8,1) > 1/2 and C_(8,2) > 1/2
MW5candidate2026-09-03
||(M_(Z₁0) chi₀,1,4,7)^((9))||ᵢnf / ||chi^((9))||ᵢnf = 139/270 > 1/2, hence C_(9,inf) > 1/2
MW6routine2026-09-03
The same Z₁0 set keeps the l¹ ratio above 1/2 at every k from 8 to 16
MR1candidate2026-09-03
C_(k,1) > 1/2 for EVERY k >= 8: the single set 0,1,4 in Z₇ beats 1/2 in l¹ at every order from 8 to 35, and the source's own thm:ckpgeq covers k >= 36
MR2candidate2026-09-03
C_(k,2) > 1/2 for EVERY k >= 8: 0,1,4,7 in Z₁0 at k = 8, 0,1,4 in Z₇ at every order from 9 to 22, the source's thm:ckpgeq from k >= 23
MR3candidate2026-09-03
C_(k,inf) > 1/2 for EVERY k >= 9: 0,1,4,7 in Z₁0 at k = 9, 0,3,8,9,13,14,15,16 in Z₂1 at k = 10 (ratio 35726/71379), 0,1,4 in Z₇ at every order from 11 to 35, the source's thm:ckpgeq from k >= 36
ME1known data2026-09-03
Kernel certification of the source's reported experiment: for every S in Z_N with N <= 15 and every k in 2,...,6, the circle ratio is <= 1/2 in l¹ and in lⁱnf
ME2candidate2026-09-03
The same sweep at k = 7, one order beyond the range the source reports: no subset of any circle of length <= 15 beats 1/2 in l¹ or lⁱnf
MN1routine2026-09-03
Negative control: both witnesses fail at k = 7 in all three norms – the crossing at k = 8 is real, not an artefact of the encoding
MN2routine2026-09-03
Negative control: the l¹ witness 0,1,4 in Z₇ does not beat 1/2 in lⁱnf at k = 8 (its lⁱnf ratio there is 7/15)
MN3routine2026-09-03
Negative control: the Z₇ ratio at k = 8 does not exceed 27/50
MN4routine2026-09-03
Negative control: the neighbouring set 0,1,3 in Z₇ beats 1/2 at no k <= 16
MD1measurement2026-09-03
Measurement: the exhaustive circle search and its price – 27.7 microseconds per necklace class at N = 24, scaling as (N/24)², completed to N = 28; N = 34 parked at 7.9 CPU-h
This ledger entry is reported in prose and is not bound to a Lean theorem.Provenance
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- Machina Mathematica
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- Korea Superintelligence Labs
- Source context
- For f: ℤ → ℝ let M f(n) = sup_(I ∋ n) (1/#I) ∑_(i∈I) |f(i)| be the uncentered discrete Hardy–Littlewood maximal operator (I a discrete interval), and let f^((k)) be the k-th forward difference, f^((k+1))(n) = f^((k))(n+1) − f^((k))(n). Liao–Madrid–Palsson–Weigt, *Sharp higher order regularity of discrete maximal functions*, arXiv:2607.10753v2 (math.CA), define
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
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