Infinite modulus between a compact real set and its real complement
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Original abstract
Let E⊂[0,1] be compact and let F=ℝ∖E be its complement within the real axis. We prove that positive planar 2-modulus of the family joining E to {|z|=2} implies infinite modulus of the family joining E to F. Admissibility is imposed on every curve for pointwise extended Borel densities. A weighted-distance potential and a critical trace estimate first force any hypothetical finite-energy separator to have ℒ¹(E)=0; a separate small-circle argument then treats every such null set, including null sets of positive conformal capacity. The result is stated under explicit conventions for the question recorded as Hayman–Lingham Problem 7.58.
The idea in plain English
Planar 2-modulus measures the least squared-area cost of a nonnegative density whose line integral is at least one along every curve in a family. This paper compares curves from a compact subset of the real axis to a surrounding circle with curves from that set to its complement on the real axis.
If the first family has positive modulus, the second has infinite modulus. A hypothetical finite-energy density separating the two real sets first forces the compact set to have zero length. A separate small-circle argument then treats the remaining null sets, including ones of positive conformal capacity.
The pointwise and every-curve conventions matter. Infinite density at an endpoint alone adds no line-integral cost, and the proof does not discard exceptional curves or exceptional starting points.
The precise result
For every compact E⊂[0,1], let F=ℝ∖E be its complement within the entire real axis, and let S₂={z∈ℂ:|z|=2}. Then Mod₂Γ(E,S₂)>0 implies Mod₂Γ(E,F)=∞. The ambient domain for both curve families is the whole plane.
A connector is a continuous curve with its stated endpoints, locally rectifiable on the open parameter interval. Its integral is the increasing limit on compact interior subcurves. Arclength counts multiplicity; constant portions and isolated endpoints have zero arclength measure. The whole connector may have infinite Euclidean length. Intermediate contacts with either plate or the real axis are allowed, including crossings and revisits.
Admissible densities are actual pointwise nonnegative extended Borel functions ρ:ℝ²→[0,∞], required to satisfy ∫_γρ ds≥1 for every connector. Modulus is inf ∫_ℝ²ρ² dA over those densities, with inf ∅=∞. Values may be infinite on area-null sets; replacing a density only almost everywhere need not preserve this every-curve condition.
The complete theorem also holds when both families consistently use globally rectifiable connectors. It does not mix the two conventions or remove a modulus-zero family of exceptional curves. The statement fixes explicit conventions for the geometric implication recorded in Hayman–Lingham Problem 7.58.
The contrapositive is stronger than a length conclusion: a finite-energy density admissible for Γ(E,F) produces q∈L²(ℝ²) with ∫_γq ds=∞ for every γ∈Γ(E,S₂). Therefore q/n is admissible for the entire outer family for every positive integer n, and its energy tends to zero. No extremal density is required.
Empty E has zero outer modulus. Positive-length compact sets without intervals are covered by the trace argument; null compact sets of positive conformal capacity are covered by the separate circle argument. The positive-outer-modulus hypothesis is retained, so the theorem does not assert infinite complement modulus for every compact set without that hypothesis.
Why the result holds
On a compact rectangle, dyadic open covers give a lower semicontinuous g≥max{ρ,1} with finite squared integral and pointwise domination everywhere, including infinite values on null sets. Weighted-length lower semicontinuity and the bound g≥1 give compactness of bounded-cost paths. The resulting truncated weighted-distance potential u is an actual Borel function, with u=0 on E and u=1 on F inside the rectangle.
An all-path upper-gradient inequality makes the actual u absolutely continuous on almost every coordinate line and a Sobolev W¹,² function. A compact cutoff of 1−u has actual vertical trace 1_E almost everywhere on the axis. No representative is arbitrarily reset on the axis, and no capacity conclusion is drawn from the exceptional vertical lines.
A Fourier trace estimate and mollification give a finite critical half-order difference seminorm for 1_E, without dividing by the frequency at zero. Telescoping translations of an indicator function give a lower bound proportional to length(E) times ∫₀¹ dh/h. Finiteness therefore forces length(E)=0. This classical rigidity step does not imply zero conformal capacity.
For the remaining null sets, set q=ρ+1_{B(0,3)}. Any finite-q-cost connector to S₂, stopped at its first hit, has finite total variation even at its initial endpoint and can be parametrized by arclength. Fix each actual e∈E separately. Polar integration selects radii r_j↓0 with e+r_j∈F and r_j∫_{|z−e|=r_j}q² ds→0. This is a liminf subsequence, not a claim of a full radial limit or a uniform endpoint selection.
The curve's first-exit initial cost tends to zero, and a circular arc to e+r_j adds cost at most √(2πr_j∫_{|z−e|=r_j}q² ds), also tending to zero. Their concatenation is an actual globally rectifiable E-to-F connector of cost below one, contradicting admissibility. Returns, retracing, winding and intermediate plate contacts do not invalidate this construction. It works at every endpoint and leaves every outer curve in the final scaling argument.
Scope and limitations
The complement is real, not planar. The result uses whole-plane planar 2-modulus, pointwise extended Borel densities and admissibility on every curve under the stated local-off-endpoint convention or the separately consistent global convention. Other domains, exponents, plate-avoiding interior requirements or density-equivalence-class-only conventions are not asserted.
Zero length and zero conformal capacity are different. The trace step alone does not settle the null positive-capacity case; the endpoint-by-endpoint circle proof is essential. A singleton is consistent with zero outer modulus, and infinity placed only at an endpoint is not a separating line-integral barrier.
The argument uses classical measure theory, weighted-distance/upper-gradient ideas, Arzelà–Ascoli, Sobolev mollification, Plancherel and critical trace rigidity. It makes no new-mechanism claim. Hesse's compact separated-plate theory is background; no capacity identity for the touching real complement is imported.
The question is attributed to A. A. Gonchar, communicated by M. Vuorinen, in the 2018 Hayman–Lingham collection. The full 2019 Springer problem wording and the mathematical contents of the 1982 Goldstein–Jackson item remain unauthenticated. Its programme title does not establish an earlier resolution or its exact relevance. A historical status statement or update-list silence does not establish current worldwide openness.
Historical first priority, independent discovery and current worldwide open status remain undetermined. An indirect deduction assembled from older theory is not evidence that an earlier author printed this exact resolution. The public statement concerns the manuscript's explicit conventions and does not claim to settle every interpretation of unspecified source terminology.
Status and AI use
Unrefereed preprint, version 1.0. Developed with extensive AI assistance; no external human peer review or formal proof certification.
AI tools were used extensively in research, proof development, literature work, writing and internal adversarial checking. This is an unrefereed preprint without external human peer review, journal acceptance or formal proof-assistant certification. The named author is responsible for the work. Internal checking and finite controls do not replace the analytic proof or certify historical priority.
Manuscript and declared publication date: 17 October 2026; version 1.0.
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Cite this paper
Alec Kriebel. (2026). Infinite modulus between a compact real set and its real complement (Version 1.0) [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.23441617
Background and prior work
- W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (New Edition), arXiv:1809.07200v2 (2018): Problem 7.58, printed page 178, records the compact-real-set and real-complement geometry and credits Gonchar via Vuorinen. Its historical update does not certify present openness.
- E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, arXiv:1104.4345v3 (2011): Propositions 3.4 and 3.8 give classical Fourier/Gagliardo and trace background. The note proves the special flat trace estimate it needs directly.
- H. Brezis, How to recognize constant functions. Connections with Sobolev spaces (2002): Corollaries 1–2 and Remark 3 supply classical critical rigidity context. Remark 3 also credits Hardt–Kinderlehrer–Lin and their Wiener attribution; that predecessor credit is mediated through Brezis.
- J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Math. 181 (1998), 1–61: Established all-curve modulus, pointwise-majorant, weighted-potential and upper-gradient background. The present mechanisms are credited as classical.
- J. Hesse, A p-extremal length and p-capacity equality, Ark. Mat. 13 (1975), 131–144: Theorem 5.5 concerns compact separated-plate context. It is not extended here to the touching, noncompact real complement; the direct proof avoids that step.