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Arithmetic-progression gap sets in Cantor sets

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Pith's one-line read For every middle-ε Cantor set with 3−2√2 < ε ≤ 1/3, the longest arithmetic progression has exactly four terms, so the maximal length drops immediately from six at the boundary.

desk verdict The four-term maximal AP length result is clean and new; the HKY-based positive interval theorems are promising but rest on an under-verified initial-stage reduction whose margin is uncomfortably tight. read the letter →

arxiv 2608.14998 v1 pith:2DAB2C6F submitted 2026-08-15 math.CA

Samantha Sandberg-Clark , Krystal Taylor , Alexia Yavicoli This is my paper · ORCID
classification math.CA MSC 28A8011B2528A78
keywords arithmetic-progressiongapsetsCantormiddle-εsetNewhousethicknessHunt-Kan-Yorkeconstructionblackoutintervalsscalefractalgeometry
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Arithmetic progressions inside a compact set are usually studied one configuration at a time; this paper quantifies the full set of step sizes at which they occur. Its main exact result is that in the middle-ε Cantor set $C_\varepsilon$, the longest arithmetic progression has exactly four terms for every $3-2\sqrt2<\varepsilon\le 1/3$, so the maximal length drops immediately from six (the known value at $\varepsilon=3-2\sqrt2$) to four. The paper also proves two complementary statements about the gap set $G^k_{\mathrm{AP}}(C)$: recursive outer bounds and explicit blackout intervals, including the result that $G^3_{\mathrm{AP}}(C_\varepsilon)$ contains no interval $(0,r)$ whenever $\varepsilon>(4-\sqrt{13})/3$. On the positive side, it shows that sufficiently thick Cantor sets contain every sufficiently small common difference: with largest bounded gap at most $0.067d$ and thickness at least $6.96268\ldots$, every $s\in(0,0.435d]$ occurs as a three-term progression step, with analogues for four-term progressions and asymmetric three-point patterns. These uniform scale intervals are what a dimension-only criterion cannot supply.

What carries the argument

Two mechanisms carry the argument. First, the first-splitting decomposition: every arithmetic progression in an attractor of an iterated function system either lies in one construction interval or, after finitely many inverse branches, splits between two first-level pieces, so the full gap set is contained in a scaled union of the first-split spectrum; for $C_\varepsilon$ this gives $G^k_{\mathrm{AP}}(C_\varepsilon)\subseteq \bigcup_{n\ge0}\lambda^n[\varepsilon,c_k(\varepsilon)]$ and the blackout criterion $\lambda c_k<\varepsilon$. Second, the quantitative intersection mechanism: the paper refines the Hunt–Kan–Yorke construction from [12] by extracting an explicit first-stage lower bound $\tilde{\psi}_t(\tau_1,\tau_2)$ on distances between the enlarged gaps, which yields an explicit minimum thickness $\tilde{\varphi}_t(\tau)$ for the intersection set $K\subseteq C_1\cap C_2$; then the Newhouse Gap Lemma—interleaved compact sets with thickness product at least 1 intersect—is applied to $K$ and a third (or translated fourth) copy of $C$. This combination converts a qualitative positive-thickness statement into the uniform intervals $(0,0.435d]$, $(0,0.87d]$, and $(0,0.3d]$.

What would settle it

Build a self-similar Cantor set with largest bounded gap exactly $0.067d$ and thickness exactly $6.96268\ldots$, and check computationally whether $(C-s)\cap C\cap(C+s)$ is empty for some $s\in(0,0.435d]$; one empty intersection would disprove Theorem 4.1. Independently, search for a five-term progression in $C_\varepsilon$ with $\varepsilon$ just above $3-2\sqrt2$; finding one would disprove Theorem 2.4.

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Extended reading notes

Core claim

The central claim is that the scale set of arithmetic progressions in a Cantor set is governed by two opposing, explicit mechanisms, neither of which is visible from Hausdorff dimension alone. The exclusion mechanism: in the self-similar family $C_\varepsilon$, pulling a progression back through the construction until it first splits yields recursive restrictions, and for $\varepsilon\in(3-2\sqrt2,1/3]$ these restrictions force $L_{\mathrm{AP}}(C_\varepsilon)=4$ exactly; combined with the endpoint value $L_{\mathrm{AP}}(C_{3-2\sqrt2})=6$, the drop to the right is immediate. The inclusion mechanism: a two-Cantor-set intersection construction from [12], read quantitatively at its first stage, produces a compact intersection set $K$ with an explicit positive thickness bound; one more application of the Newhouse Gap Lemma turns the existence of $K$ into the statement that all small steps lie in the gap set. The paper states these as Theorem 2.4, Theorem 3.9, Theorem 4.1, Corollary 4.2, and Theorem 4.3, with the interval endpoints $0.435d$, $0.87d$, and $0.3d$ and the blackout constants $3-2\sqrt2$ and $(4-\sqrt{13})/3$.

Load-bearing premise

The positive interval theorems stand on the unproved distance bound for the enlarged-gap construction taken from the black-box Lemma 4 of [12]; if that bound is false, Theorems 4.1, 4.2, and 4.3 lose their support.

Editorial extensions

If this is right

  • In the middle-$\varepsilon$ family, the maximal progression length is now known exactly on $3-2\sqrt2<\varepsilon\le 1/3$: four terms, and the jump from the endpoint value six is one-step.
  • No interval of common differences near zero exists in $G^3_{\mathrm{AP}}(C_\varepsilon)$ for $\varepsilon>(4-\sqrt{13})/3$; any future search for a step-size interval in this family must go below this threshold.
  • Every Cantor set that meets the gap and thickness hypotheses contains every three-term progression step up to 43.5% of its diameter, and every asymmetric $\{0,\theta,1\}$ pattern up to 87% of its diameter.
  • Guaranteeing four-term progressions costs more thickness: $\tau\ge 32.83333\ldots$, and the guaranteed step interval shrinks to $(0,0.3d]$; the constants quantify a trade-off between pattern length and scale range.
  • For the middle-third Cantor set, the equality $G^3_{\mathrm{AP}}=G^4_{\mathrm{AP}}=\{3^{-n}:n\in\mathbb{N}\}$ appears as a boundary case of the recursive containment, placing the earlier qualitative bounds of [8] as the endpoint.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Going beyond the paper, the exact 6-to-4 jump suggests that $\varepsilon\mapsto L_{\mathrm{AP}}(C_\varepsilon)$ is not a gentle staircase: the maximal length can fall by two at a single parameter, so any conjecture predicting unit steps or monotonicity in that range needs revision.
  • The paper's own conclusion points to computing the first-split spectrum $B_\varepsilon$ exactly or a level-three outer approximation as the route to lower blackout thresholds; that is a concrete next computation.
  • The monotonicity claims for $\tilde{\psi}_t$ and $\tilde{\varphi}_t$ in $\tau$ imply that the constants $0.067$, $0.435$, $0.87$, and $0.3$ are not optimized; a different choice of anchor $t$ in the construction could shift the interval endpoints.
  • The flexible Proposition 4.12, stated for arbitrary compact sets $C_1,C_2,C_3$, should carry the same explicit-scale mechanism to other $k$-point configurations and to higher dimensions, since the argument is written for general interleaved sets rather than for arithmetic progressions specifically.
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Formalized claims in Lean

  1. Claim #1: The central claim is that the scale set of arithmetic progressions in a Cantor set is governed by two opposing, explicit mechanisms, neither of which is visible from Hausdorff dimension alone. The exclusion mechanism: in the self-similar family $C_\varepsilon$, pulling a progression back through the construction until it first splits yields recursive restrictions, and for $\varepsilon\in(3-2\sqrt2

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper studies the sets of common differences of arithmetic progressions and, more generally, the scale sets of finite point patterns in compact subsets of the real line, with special attention to middle-epsilon Cantor sets C_epsilon. On the exclusion side, the paper proves that for 3-2*sqrt(2) < epsilon <= 1/3 the longest arithmetic progression in C_epsilon has exactly four terms (Theorem 2.4), complementing the known endpoint value six at epsilon = 3-2*sqrt(2); it also develops a recursive first-splitting framework and a level-two refinement that yields explicit blackout intervals for G^3_AP(C_epsilon), improving the threshold to epsilon* = (4 - sqrt(13))/3 (Theorem 3.9). On the positive side, the paper extracts explicit bounds from the Hunt-Kan-Yorke intersection construction and combines them with the Newhouse Gap Lemma to show that sufficiently thick Cantor sets have intervals of admissible common differences: if the largest bounded gap is at most 0.067d and the thickness is at least 6.96268..., then (0, 0.435d] is contained in G^3_AP(C) (Theorem 4.1), with analogues for asymmetric three-point patterns (Corollary 4.2) and for four-term progressions (Theorem 4.3).

Significance. If the results are correct, this is a solid contribution to the quantitative study of patterns in fractal sets. Theorem 2.4 gives a sharp discontinuity statement for maximal arithmetic-progression length in the middle-epsilon family, and the level-two computation of Proposition 3.8, with its row-by-row certificate in Appendix A, is a concrete and reproducible example of how product geometry improves first-level estimates. The positive results are notable because they convert the qualitative Hunt-Kan-Yorke theorem into explicit parameter ranges with numerical constants, and because the extension from equal-thickness anchors to all larger thicknesses is made explicit. The paper is largely self-contained and is careful to attribute the underlying HKY mechanism and the Newhouse Gap Lemma. However, the main positive theorems currently rest on a lemma whose proof is incomplete for the parameter values actually used in the applications, and one monotonicity claim contains an algebraic error; both issues are repairable without changing the announced results.

major comments (3)
  1. [Section 4.2, Claim 4.14] The proof of Lemma 4.9 establishes the key lower bound only for t sufficiently close to 0. After normalizing d(G,H)=1, the proof shows that 1 - sigma_1 lambda_1 - sigma_2 lambda_2 is positive at t=0 and then invokes continuity. The applications in Theorems 4.1 and 4.3 use the fixed values t* = 0.55 and t* = 2.5, which are not covered by that argument. The gap is repairable because the estimates (4.4)-(4.6) do not require small t: one should either prove Lemma 4.9 on the whole range 0 < t < t_max using the explicit positivity statement in Remark 4.6, or, in the symmetric applications, verify directly that the chosen t* lies in the positivity range (tau*^2 - 2 tau* - 1)/(3 tau* + 1) and state explicitly that the chain preceding the positivity check applies at that t*. As written, however, the proof of Theorem 4.1's interval conclusion is not fully supported by the lemma actually proved.
  2. [Section 4.2, Claim 4.14] The derivative in Claim 4.14 is algebraically incorrect. For psi_t(tau) as defined in equation (4.11), differentiation gives numerator 2(1+t) tau^2 - t(t+2) tau - (1+2t)(2t^2+3t+1), not 2(1+t) tau^2 - t(4t+5) tau + 2(1+t)(3t^2+3t+1) as printed. The conclusion of the claim is nevertheless true on the relevant domain: the corrected numerator is positive at tau = 2t+1, where it equals 2t^3+3t^2+3t+1, and its derivative is positive for tau > 2t+1. The proof should be corrected because the printed computation does not establish the monotonicity used to pass from the anchor tau* to all tau >= tau*.
  3. [Section 4.2, Proposition 4.13] In the proof of Proposition 4.13, Lemma 1.4 is applied with the non-strict inequality s <= diam(K), while Lemma 1.4 requires 0 < u < diam(K) for the sets K and K+u to be interleaved. When equality s = diam(K) holds, interleaving is not guaranteed. This is easily repaired: either prove the interval claim for s strictly below the endpoint and use closedness of G^4_AP(C) to pass to the endpoint, or show that the construction gives strict inequality under the stated hypotheses. As written, the endpoint case is not justified.
minor comments (5)
  1. [Proposition 4.12 proof] In the proof of Proposition 4.12, the sentence 'condition (i) guarantees C subset Q subset conv(C3)' appears to contain a typo; it should likely read 'conv(K) subset Q subset conv(C3)' or an equivalent statement about K.
  2. [Proposition 4.13 proof] The final sentence of the proof says the four points form a progression 'with common differences'; this should be 'with common difference s'.
  3. [Theorem 3.1 proof] The phrase 'whichgivesthethirdupperbound' is missing spaces and should read 'which gives the third upper bound'.
  4. [Lemma 4.9 proof] The text 'R_{i+1} = max(R_i - x_i - y_i, 0) for I >= 0' uses a capital 'I'; this should be the index 'i'.
  5. [Remark 4.6] The notation in Remark 4.6 is confusing: the expansion is written as at^2+bt+c but the coefficients are named A, B, C; the authors should align the notation or rename the coefficients.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation; central claims are built on external HKY and Newhouse Gap Lemma, with only contextual self-citations.

full rationale

The derivation chain is self-contained against external results. Theorem 2.4 follows from Proposition 2.1, Theorem 2.2, and Lemma 2.3, which are proved from the self-similar structure of the middle-epsilon Cantor set and the Newhouse Gap Lemma; no equation in Section 2 is assumed as its own conclusion. The blackout results (Theorems 3.1 and 3.9) are combinatorial first-splitting bounds with explicit endpoint computations, independently reproduced in Appendix A. The positive interval theorems (4.1-4.3) rest on the Hunt-Kan-Yorke construction [12] and its Lemma 4, which is explicitly used as a black box in Lemma 4.8, together with the external Newhouse Gap Lemma [13]. These are prior external results, not the authors' own unverified claims. The paper's self-citations ([4], [14], [16], [17]) are contextual background or existence-oriented statements and are not load-bearing premises in any proof. The 'anchor values' tau_star=6.96 and t_star=0.55 are parameter choices verified by monotonicity claims, not fitted data; the constants in Theorem 4.1 are computed from explicit formulas rather than calibrated to the conclusion. The authors themselves flag the unverified N=1 possibility and the black-box use of Lemma 4.8, which are correctness risks but not circularity. No prediction is defined in terms of an input, and no self-citation supplies the forcing step. Score 2 reflects only the presence of minor non-load-bearing self-citations; there are no circular steps.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

No free parameters are fitted to data; the only hand-chosen numbers are proof anchors that the explicit interval endpoints depend on. The listed axioms are external theorems the paper relies on without reproving.

free parameters (1)
  • HKY anchor constants = t=0.55, tau=6.96, |G|=0.067 for Theorem 4.1; t=2.5, tau=32.8 for Theorem 4.3
    Hand-chosen proof constants satisfying the inequalities in (4.15), phi >= 1/tau, and the HKY region-IV conditions. They are not empirical fits, but the explicit interval endpoints (0.435, 0.87, 0.30) depend on these choices.
assumptions (3)
  • standard math Newhouse Gap Lemma (Prop 1.3)
    External result from [13] used to force intersections from interleaving and thickness product; restated in Section 1.
  • domain assumption Hunt-Kan-Yorke construction and Lemma 4 (Lemma 4.8)
    Black-box prior theorem from [12] guaranteeing the gap system (K_n) and distance bounds used to prove Lemma 4.9 and the positive interval theorems.
  • standard math Tian-Lou-Shang endpoint L=6 at epsilon=3-2*sqrt(2)
    Cited exact value (0.4) from [11], used to frame the drop to four; not used inside the proof of Theorem 2.4.

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Pith. "Pith review of Arithmetic-progression gap sets in Cantor sets." pith.science (2026). https://pith.science/paper/2DAB2C6F

@misc{pith2026260814998,
  author       = {Pith},
  title        = {Pith review of: Arithmetic-progression gap sets in Cantor sets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2DAB2C6F}},
  note         = {Machine review of arXiv:2608.14998}
}
abstract

We address the question of which common differences can arise in arithmetic progressions contained in fractal sets. For a compact set $C\subset\mathbb R$, we investigate not only whether arithmetic progressions occur in $C$, but the full collection of their common differences. More generally, for a finite pattern $P$, we study the set of scales at which affine copies of $P$ appear in $C$. For affine self-similar sets satisfying strong separation, we obtain explicit restrictions on admissible common differences. Specializing to middle-$\varepsilon$ Cantor sets, we prove that the longest arithmetic progression has length four whenever $3-2\sqrt2<\varepsilon\le 1/3$, showing that the maximal progression length drops immediately from six at the critical parameter $\varepsilon=3-2\sqrt2$. We further develop recursive bounds for the sets of admissible common differences and derive explicit blackout intervals, namely ranges of scales for which arithmetic progressions cannot occur. On the positive side, sufficiently thick Cantor sets exhibit the opposite behavior. Combining a refinement of the Hunt-Kan-Yorke construction with the Newhouse Gap Lemma, we prove that every sufficiently small common difference occurs in a three-term arithmetic progression. In particular, if the largest bounded gap of a Cantor set is at most $0.067 diam(C)$ and its thickness is at least $6.96268\ldots$, then every common difference in $(0,0.435 diam(C)]$ occurs in a three-term arithmetic progression contained in $C$. Analogous interval results are obtained for four-term arithmetic progressions and asymmetric three-point patterns.

Figures

Figures reproduced from arXiv: 2608.14998 by the authors.

Figure 1
Figure 1. Principal conclusions for the middle-ε family. A bar whose left end￾point is 0 records a conclusion valid for every smaller positive ε. The endpoints shown in the positive-interval results are conservative rounded constants; strict and non-strict parameter endpoints are as stated in the theorems. Definition 1.1. (bridge thickness) Let C ⊂ R be a compact set, and let (Gn) be the bounded connected components of the co… view at source ↗
Figure 2
Figure 2. In §4.1.2, we review how to construct the set K in C1 ∩ C2. In [12], this is accomplished by first taking the largest gap G of C1, C2 and then expanding it to absorb nearby gaps that are within distance t|G| for some arbitrary 0 < t < τ1τ2−1 τ1+τ2+2 . They then repeat this process until no gaps remain. This creates a new system of gaps (Kn), which are dependent on the number t; note that smaller values of t make for… view at source ↗
Figure 2
Figure 2. The intersection of two interleaved compact sets with thicknesses τ1 and τ2: can be empty for (τ1, τ2) ∈ I; is nonempty for (τ1, τ2) ∈ II; contains a set of positive thickness for (τ1, τ2) ∈ III; and contains a set with an explicit minimum thickness in IV . These inequalities come from the original argument in Hunt-Kan-Yorke [12], and all thickness pairs (τ1, τ2) in regions III or IV guarantee the intersection of C1… view at source ↗
Figures from the paper (3 more)
Figure 3
Figure 3. Figure 3: Cases in the construction of I∗ and J∗. one of the bordering gaps being bounded. Pick a chunk P, and say P is bordered by Km and Kn with m > n and m ≥ 2. Then |P| d(P, K \ P) = d(Km, Kn) min(|Km|, |Kn|) ≥ d(Km, Kn) |Km| . Recalling that thickness is defined as τ (K) = …
Figure 4
Figure 4. Figure 4: The construction of K0. not extend K0 any farther rightward, we stop the construction. If not, we then add to K0 the rightmost Jl which is within t times its length of Im and is at most as large as Jn (see [PITH_FULL_IMAGE:figures/full_fig_p022_4.png]
Figure 5
Figure 5. Figure 5: The gaps Gi and Hi . for i ≠ j. If d(G, H) = 0, the inequality to be proven is trivial. Otherwise, let us normalize d(G, H) to one, and assume G lies to the left of H. Let G0 = G and H0 = H. Let G1 be the 1-gap of Kn adjacent to H0 on its left, and let H1 be the 2-gap…

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Works this paper leans on

17 extracted references · 16 canonical work pages

  1. [1]

    K. J. Falconer,On the Hausdorff dimensions of distance sets, Mathematika32(1985), no. 2, 206–212

    work page 1985

  2. [2]

    Mattila and P

    P. Mattila and P. Sjölin,Regularity of distance measures and sets, Math. Nachr.204(1999), 157–162

    work page 1999

  3. [3]

    Keleti,Construction of one-dimensional subsets of the reals not containing similar copies of given patterns, Anal

    T. Keleti,Construction of one-dimensional subsets of the reals not containing similar copies of given patterns, Anal. PDE1(2008), no. 1, 29–33

    work page 2008

  4. [4]

    Yavicoli,Large sets avoiding linear patterns, Proc

    A. Yavicoli,Large sets avoiding linear patterns, Proc. Amer. Math. Soc.149(2021), no. 10, 4057–4066

    2021

  5. [5]

    Łaba and M

    I. Łaba and M. Pramanik,Arithmetic progressions in sets of fractional dimension, Geom. Funct. Anal. 19(2009), no. 2, 429–456

    work page 2009

  6. [6]

    Carnovale and S

    M. Carnovale and S. Senger,On the number of3APs in fractal sets, preprint, arXiv:2602.03029 (2026)

    work page arXiv 2026

  7. [7]

    Shmerkin,Salem sets with no arithmetic progressions, Int

    P. Shmerkin,Salem sets with no arithmetic progressions, Int. Math. Res. Not. IMRN (2017), no. 7, 1929–1941

    work page 2017

  8. [8]

    Broderick, L

    R. Broderick, L. Fishman, and D. Simmons,Quantitative results using variants of Schmidt’s game: dimension bounds, arithmetic progressions, and more, Acta Arith.188(2019), no. 3, 289–316. ARITHMETIC-PROGRESSION GAP SETS 33

    work page 2019

Show all 17 references
  1. [9]

    Chaika,Arithmetic progressions in middleNth Cantor sets, arXiv:1703.08998 (2017)

    J. Chaika,Arithmetic progressions in middleNth Cantor sets, arXiv:1703.08998 (2017)

    2017 arXiv

  2. [10]

    L. Xi, K. Jiang, and Q. Pei,Arithmetic progressions in self-similar sets, Front. Math. China14(2019), no. 5, 957–966

    2019

  3. [11]

    Z. Tian, M. Lou, and L. Shang,Arithmetic progressions and thickness properties in self-similar Cantor sets, Discrete Contin. Dyn. Syst. Ser. S25(2026), 160–175

    2026

  4. [12]

    B. R. Hunt, I. Kan, and J. A. Yorke,When Cantor sets intersect thickly, Trans. Amer. Math. Soc.339 (1993), no. 2, 869–888

    1993

  5. [13]

    S. E. Newhouse,Nondensity of axiom A(a)on S2, in Global Analysis (Proc. Sympos. Pure Math., Vol. XIV), Amer. Math. Soc., Providence, RI, 1970, pp. 191–202

    1970

  6. [14]

    Taylor and S

    K. Taylor and S. Sandberg-Clark,Triangles in the plane and arithmetic progressions in thick compact subsets ofR d, Canad. J. Math.78(2026), no. 3, 946–986

    2026

  7. [15]

    R. F. Williams,How big is the intersection of two thick Cantor sets?, in Continuum Theory and Dynamical Systems (Arcata, CA, 1989), Contemp. Math.117, Amer. Math. Soc., Providence, RI, 1991, pp. 163–175

    1989

  8. [16]

    Yavicoli,Patterns in thick compact sets, Israel J

    A. Yavicoli,Patterns in thick compact sets, Israel J. Math.244(2021), no. 1, 95–126

    2021

  9. [17]

    Yavicoli,A survey on Newhouse thickness, fractal intersections and patterns, Math

    A. Yavicoli,A survey on Newhouse thickness, fractal intersections and patterns, Math. Comput. Appl. 27(2022), no. 6, Art. 111. Department of Mathematical Sciences, United States Military Academy at West Point Email address:samantha.sandbergclark@westpoint.edu Department of Mat...

    2022

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