bibkey: dukesgaede2023erdosdeep authors: Peter J. Dukes, Tao Gaede year: 2023 title: “Families of Modular Arithmetic Progressions with an Interval of Distance Multiplicities” doi: 10.48550/arXiv.2208.05527 url: https://math.colgate.edu/~integers/x25/x25.pdf claim: “Conjecture 1 proposes that the length triples of Erdos-deep families of three modular arithmetic progressions are exactly two infinite families and thirteen sporadic triples.” strata_touched:

D5/S3/Combinatorics/ErdosDeepTripleClassificationRefutation license: citation-only triage: anchor

Dukes–Gaede modular arithmetic progressions with an interval of distance multiplicities

Verified locator

URL: https://math.colgate.edu/~integers/x25/x25.pdf

DOI: 10.48550/arXiv.2208.05527

Version: INTEGERS 23 (2023), Article #A25, by Peter J. Dukes and Tao Gaede. The arXiv record 2208.05527 carries only v1 (10 August 2022) and lists no later version. The same conjecture appears as Conjecture 4.5 of Tao Gaede’s University of Victoria master’s thesis Erdős-Deep Families of Arithmetic Progressions, which states that its Chapters 3 and 4 are the basis of this article; the thesis version carries one hypothesis the article does not display, recorded below.

Definitions

Section 1 works in Z_n with |x|_n := min(x, −x), each of ±x reduced in {0, …, n−1}, and sets dist(x, y) = |x − y|_n. For a family F = {A_1, …, A_s} of subsets, ΔF is the multiset of the distances dist(x, y) over pairs {x, y} ⊆ A_i, x ≠ y, for each i. The article defines

we say that F is Erdős-deep if the multiplicities of distances that occur in ΔF are precisely 1, 2, …, k − 1 for some integer k.

Here k is determined by k(k−1) = Σ_i k_i(k_i−1) where k_i = |A_i|. It also sets AP_n(g, k) := {0, g, 2g, …, (k−1)g} ⊂ Z_n, the modular k-term AP. Theorem 2, the classification for s = 2, fixes the conventions carried over to families of three: k_1 ≥ k_2 ≥ 3 and gcd(n, g_1, g_2) = 1. Two members of a family may coincide: the article’s own (6, 3, 3) witness is {{0,1,2,3,4,5}, {0,4,8}, {0,4,8}}.

The statement in question

Section 5 states:

Conjecture 1. An Erdős-deep family of three APs of lengths k1 ≥ k2 ≥ k3 in Z_n exists if and only if (k1, k2, k3) ∈ {(4, 4, 3), (6, 3, 3)}, each for infinitely many n, or (k1, k2, k3) ∈ {(6, 5, 3), (6, 6, 4), (6, 6, 6), (7, 7, 3), (9, 4, 3), (8, 7, 4), (8, 8, 5), (10, 6, 4), (12, 4, 4), (13, 5, 3), (13, 7, 4), (16, 5, 4), (21, 6, 4)}, each for a finite number of values of n.

Conjecture 4.5 of the thesis is the same statement written in the tuple form (k, k1, k2, k3), with the extra hypothesis 3 ≤ k3 ≤ k2 ≤ k1 ≤ ⌊n / (2·gcd(n, g1))⌋ + 1. The thesis also prints the search that produced the list: k ∈ [4, 28]; g1, g2, g3 ≤ ⌊n/2⌋; 3 ≤ k3 ≤ k2 ≤ k1 ≤ ⌊n/(2·gcd(n,g1))⌋; n ∈ [2k1, 60]; and, when gcd(g1, n) = 1, only g1 = 1 is checked, which is a valid reduction because scaling every generator by a unit permutes the distances.

Scope of the recorded answer

The conjecture is false, in both forms. In Z_27 take

A₁ = AP₂₇(1, 12) = {0,1,2,3,4,5,6,7,8,9,10,11}
A₂ = AP₂₇(12, 6) = {0, 12, 24, 9, 21, 6}
A₃ = AP₂₇(7, 5)  = {0, 7, 14, 21, 1}

with

ΔA₁ = {1:11, 2:10, 3:9, 4:8, 5:7, 6:6, 7:5, 8:4, 9:3, 10:2, 11:1}
ΔA₂ = {3:4, 6:3, 9:3, 12:5}
ΔA₃ = {1:1, 6:2, 7:4, 13:3}
ΔF  = {1:12, 2:10, 3:13, 4:8, 5:7, 6:11, 7:9, 8:4, 9:6, 10:2, 11:1, 12:5, 13:3}.

Thirteen distances occur, with multiplicities 1 through 13 each exactly once, so F is Erdős-deep with k = 14, and 14·13 = 182 = 12·11 + 6·5 + 5·4. The conventions hold: 3 ≤ 5 ≤ 6 ≤ 12, gcd(27, 1, 12, 7) = 1, and k₁ = 12 ≤ ⌊27/(2·1)⌋ + 1 = 14. But (12, 6, 5) is neither of the two infinite families nor one of the thirteen sporadic triples.

The witness lies inside the search the thesis describes: n = 27 ≤ 60, g₁ = 1, k₁ = 12 ≤ ⌊27/2⌋ = 13, and k = 14 ∈ [4, 28].

Further witnesses and a scope note

Searching n ≤ 84 and k ≤ 28 under the article’s conventions reproduces all fifteen of its triples and yields twenty-seven in total. Eleven of the twelve extra triples also satisfy the thesis bound on k₁: (9,9,4), (10,9,5), (10,10,6), (11,10,9), (11,11,5), (12,6,5), (12,9,3), (12,11,6), (18,6,3), (20,5,5), (25,6,5). A second witness that can be checked by hand is n = 50, (20,5,5), g = (1,10,10): there A₁ = {0,…,19} gives each distance d ≤ 19 multiplicity 20 − d, while A₂ = A₃ = {0,10,20,30,40} each give 10 five times and 20 five times, so the multiplicities are 1 through 20.

Separately, the article says the geometric infinite family F = {{0,1,2,3}, {0,3,6,9}, {0,1,2}} is realised for n ≥ 15. At n = 15 the distance 9 folds to |9|₁₅ = 6 and merges with the existing 6s, giving multiplicity three twice, so the family is Erdős-deep only for n ≥ 16. This does not bear on Conjecture 1, which claims only that the triple occurs for infinitely many n.

Bounded prior-resolution evidence

The journal article, the arXiv record 2208.05527 (v1 only, no later version) and the full text of Gaede’s thesis were opened. The thesis presents the statement as a conjecture and records no refutation; no later paper by either author on the topic was found. Citation-index result pages were not reachable, so this is a bounded negative finding and no worldwide priority claim is made.

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