Arnold's geometric theory of Frobenius numbers (realizators, D-domains, saws, sails) reduced the conductor N(a,b,c) = F(a,b,c)+1 of the semigroup generated by a, b, c to a continued-fraction-geometric extremum, but left his own experimental observations unexplained: conductors of related triples (a,b,c) and (a,b+1,c-1) with a+b+c = 97 are linked by exact ratios 2, 2, 2, 4, and whole one-parameter families of triples have constant conductor. Later work partially resolves the phenomenon: Fel explains the constant families through symmetric numerical semigroups and a universal Frobenius number, and Aguiló-Gost constructs infinite linked families with conductor ratios 1 and 3 via L-shaped Apéry tiles. Still open, however, are the ratio-2 regime (no infinite family is known; in Aguiló-Gost's class of pairs where both semigroups are 3-minimally generated, bounded searches found none), any complete characterization of when links occur, and Arnold's actual question: what do linked triples share in their continued-fraction/sail data? The problem asks for a provable geometric mechanism in the sail/Apéry-tile data that explains these conductor links — Arnold's actual question — and such a mechanism must be consistent with the series he displayed.

Background

Fix positive integers a, b, c with gcd(a,b,c) = 1 (Arnold's standing assumption: the three numbers have no common divisor greater than 1). The set S = ⟨a,b,c⟩ = {ax+by+cz : x,y,z ≥ 0 integers} is a numerical semigroup: it contains all sufficiently large integers. The Frobenius number F(S) is the largest integer not in S, and the conductor is N(S) = F(S)+1, the least element of S such that every larger integer also belongs to S. Arnold writes N(a,b,c) for the conductor; all numerical values in this statement follow that convention (his often-quoted values 90 and 168 are the classical Frobenius numbers 89 and 167 plus one). For two generators Sylvester's theorem gives N(a,b) = (a-1)(b-1).

For three generators an explicit formula already exists: Rødseth (1978) expressed F(a,b,c) through a finite continued-fraction computation, and Tripathi (2017) surveys the known three-variable formulae. Curtis (1990) proved that no formula from a natural class of closed-form expressions built from the generators computes the Frobenius number of every numerical semigroup, and for general n the problem is NP-hard (Ramírez Alfonsín 2005). Arnold's program was therefore not a search for a closed form but a geometric theory, developed in 1999 and presented in lectures in 2005 (published in his 2006 booklet and in the 2009 paper [1]):

  • For the linear form l(y,z) = by+cz on the closed positive quadrant, the realizator of a residue k mod a is an integer point r of the quadrant with l(r) ≡ k (mod a) and l(r) minimal among such points. The set of realizators of all a residues spans a domain D(a,b,c) (the D-domain) whose boundary is always a stepwise staircase, a Young-diagram shape (Theorem 1).
  • If L is the maximum of l over the vertices of this staircase, then N(a,b,c) = L - a + 1 (Theorem 2): the Frobenius computation becomes a geometric extremum.
  • The D-domain is the complement, in the positive quadrant, of the union of translated quadrant-shaped 'teeth' with vertices on the positive half of the lattice {(y,z) : a divides l(y,z)} — Arnold's saw (Theorem 4).
  • Under projection from 3-space along the x-axis, the same construction lives inside the plane Π = {ax+by+cz = 0} with its integer lattice: the domain attached to any of the six Weyl chambers (cones cut out by the coordinate lines) projects to D(a,b,c) (Theorem 5). Consequently all six sails — the boundary broken lines of the convex hulls of the nonzero lattice points in each chamber, i.e. the two-dimensional continued fractions of Π (Klein polygons; Karpenkov's monograph develops the general theory of such multidimensional continued fractions) — carry the same Frobenius number.

Computing thousands of conductors in 1999, Arnold recorded empirical links between conductors of related triples that his theory did not explain (all pairs below have a+b+c = 97 on both sides): N(13,32,52) = 372 = 2·186 = 2·N(13,33,51); N(5,35,57) = 224 = 2·112 = 2·N(5,34,58); N(9,43,45) = 336 = 2·168 = 2·N(9,42,46); N(4,20,73) = 216 = 4·54 = 4·N(4,19,74); the constant family N(4, 6+4k, 87-4k) = 90 for k = 0, ..., 14; and the constant family N(9, 9k±3, 88-(9k±3)) = 168 for k = 1, ..., 7 (these are the corrected ranges: Arnold's printed list excludes k = 8 from the 90-family, and the '-' branch of the 168-family in fact fails at k = 7, where the triple (9,60,28) has N = 132; the corrections follow Fel's theorems). His stated question is how the additive semigroups and the continued fractions of the members of these series are related, for triples whose Frobenius numbers are linked by the displayed formulas.

Two later frameworks are directly relevant. First, a numerical semigroup is symmetric when x is a gap of S exactly when F-x is a non-gap; Herzog (1970) proved that every three-generated symmetric semigroup has the form ⟨bc₁, bc₂, a⟩ (with suitable pairwise coprimality) with F = bc₁c₂ + ab - (bc₁ + bc₂ + a). Second, for m in S the Apéry set Ap(S,m) = {s in S : s-m not in S} determines F = max Ap(S,m) - m; for three generators and m the smallest generator, the Apéry set tiles the plane by an L-shaped tile (a minimum distance diagram L(l,h,w,y) with area lh - wy equal to the largest generator) — the same staircase object as the boundary of Arnold's D-domain. Aguiló-Gost (2011) used these tiles to compute conductors exactly, in parallel to Arnold's saw. He also formalized conductor links: P(α,k) denotes the set of pairs of semigroups ⟨α,n,m⟩, ⟨α,n-1,m+1⟩ in which the first is 2-minimally generated (two of the three listed generators already generate it, the third being redundant) and the second is 3-minimally generated (no listed generator lies in the semigroup generated by the other two), with conductor quotient N(α,n,m)/N(α,n-1,m+1) = k; Q(α,k) is defined identically except that both semigroups are 3-minimally generated. Imposing a bound m ≤ β gives the finite search classes P(α,k,β), Q(α,k,β).

Problem Statement

Let ⟨a,b,c⟩ = {ax+by+cz : x, y, z ≥ 0 integers} be the numerical semigroup generated by positive integers a, b, c with gcd(a,b,c) = 1; its Frobenius number F(a,b,c) is the largest integer not in ⟨a,b,c⟩, and its conductor N(a,b,c) = F(a,b,c) + 1 is the least element of the semigroup such that every larger integer also belongs to it (Arnold's convention — his often-quoted values 90 and 168 are the classical Frobenius numbers 89 and 167 plus one). Call two such triples (a,b,c) and (a,b',c') with b+c = b'+c' conductor-linked with ratio q when the quotient of their conductors N(a,b,c)/N(a,b',c') is the positive integer q; Arnold's examples are the case (a,b,c) vs (a,b+1,c-1).

The task is Arnold's stated open question: how are the additive semigroups and the continued fractions of the members of these series related, for triples whose Frobenius numbers are linked by the formulas he displayed? Determine what quantity or structure computable from the continued-fraction/sail data of the two triples — the D-domain staircase of realizators, the saw, the L-shaped minimum-distance-diagram tile, or the Apéry set — is provably preserved, or transformed by an exactly computable rule, across conductor-linked triples, and explains the conductor ratio. The known algebraic explanation (Fel) and combinatorial explanation (Aguiló-Gost) count only as partial progress: a successful answer must connect the ratio to the continued-fraction data itself, which is what Arnold asked for.

Because the question is scoped to the series Arnold displayed, a proposed explanation must be consistent with those observations: the ratio-2 pairs (13,32,52)/(13,33,51), (5,35,57)/(5,34,58), (9,43,45)/(9,42,46), the ratio-4 pair (4,20,73)/(4,19,74), and the two constant families with the corrected parameter ranges — k = 0, ..., 14 with no exclusion for the 90-family, and k = 1, ..., 7 on the '+' branch, k = 1, ..., 6 on the '-' branch for the 168-family.

Status of the links themselves: three of Arnold's four observed ratio pairs have ratio 2 and the fourth has ratio 4, and no infinite family of triples (a,b,c), (a,b-1,c+1) whose conductors have ratio exactly 2 or 4 has been constructed. Within Aguiló-Gost's classes, ratio-1 and ratio-3 pairs are abundant, while his bounded search found Q(α,2,100) = ∅ for α = 4, ..., 10.

Further questions from the same source, outside the scope of this task: (i) an effective sail theory of Frobenius numbers for n ≥ 4 generators — explicit formulae or provably tractable algorithms within the multidimensional-continued-fraction framework (left open by Ustinov 2012; the construction exists via Aicardi 2009 but without effectivity); (ii) a rigorous extension of Frobenius numbers to non-integral (real) arguments via infinite sails, for which only Arnold's suggestion exists. An explanation established only for a precisely delimited subclass — for example triples with a fixed smallest generator a, or Aguiló-Gost's classes P(α,k), Q(α,k) — is partial progress toward Arnold's question; it answers that question only if it covers the series he displayed.

References

  1. V. I. Arnold, Geometry of continued fractions associated with Frobenius numbers, Functional Analysis and Other Mathematics 2(2-4) (2009) 129-138. DOI 10.1007/s11853-008-0021-4, https://doi.org/10.1007/s11853-008-0021-4
  2. V. I. Arnold, Experimental'noe nablyudenie matematicheskikh faktov (Experimental observation of mathematical facts), MCCMO, Moscow, 2006, 120 pp., ISBN 978-5-94057-282-4. Lecture 4, §7. https://old.mccme.ru/free-books/dubna/via-exp.pdf
  3. F. Aguiló-Gost, Infinite families of 3-numerical semigroups with arithmetic-like links, in: Proceedings of the 3rd International Workshop on Optimal Networks Topologies (IWONT 2010), Iniciativa Digital Politècnica, 2011, pp. 11-20. Handle 2099/10341, https://upcommons.upc.edu/handle/2099/10341, https://hdl.handle.net/2099/10341
  4. L. Fel, Analytic representation in the three-dimensional Frobenius problem, Functional Analysis and Other Mathematics 2(1) (2008) 27-44. DOI 10.1007/s11853-008-0014-3, https://doi.org/10.1007/s11853-008-0014-3
  5. L. Fel, Frobenius problem for semigroups S(d1,d2,d3), Functional Analysis and Other Mathematics 1(2) (2007) 119-157. DOI 10.1007/s11853-007-0009-5, https://doi.org/10.1007/s11853-007-0009-5
  6. L. Fel, On numerical experiments with symmetric semigroups generated by three elements and their generalization, Functional Analysis and Other Mathematics 3(2) (2011) 179-192. DOI 10.1007/s11853-011-0048-9; preprint arXiv:0903.3756 (2009), https://doi.org/10.1007/s11853-011-0048-9, https://arxiv.org/abs/0903.3756
  7. F. Aicardi, On the geometry of the Frobenius problem, Functional Analysis and Other Mathematics 2(2-4) (2009) 111-127. DOI 10.1007/s11853-009-0033-8, https://doi.org/10.1007/s11853-009-0033-8
  8. A. V. Ustinov, Geometric proof of Rødseth's formula for Frobenius numbers, Proceedings of the Steklov Institute of Mathematics 276(1) (2012) 275-282. DOI 10.1134/S0081543812010245, https://doi.org/10.1134/S0081543812010245
  9. A. V. Ustinov, The solution of Arnold's problem on the weak asymptotics of Frobenius numbers with three arguments, Sbornik: Mathematics 200(4) (2009) 597-627. DOI 10.1070/SM2009v200n04ABEH004011, https://doi.org/10.1070/SM2009v200n04ABEH004011
  10. A. V. Ustinov, On the distribution of Frobenius numbers with three arguments, Izvestiya: Mathematics 74(5) (2010) 1023-1049. DOI 10.1070/IM2010v074n05ABEH002514, https://doi.org/10.1070/IM2010v074n05ABEH002514
  11. D. A. Frolenkov, The mean value of Frobenius numbers with three arguments, Izvestiya: Mathematics 76(4) (2012) 760-819. DOI 10.1070/IM2012v076n04ABEH002604, https://doi.org/10.1070/IM2012v076n04ABEH002604
  12. I. S. Vorob'ev, On the Frobenius problem for three arguments, Sbornik: Mathematics 207(6) (2016) 816-840. DOI 10.1070/SM8370, https://doi.org/10.1070/SM8370
  13. F. Curtis, On formulas for the Frobenius number of a numerical semigroup, Mathematica Scandinavica 67 (1990) 190-192. DOI 10.7146/math.scand.a-12330, https://doi.org/10.7146/math.scand.a-12330
  14. Ö. J. Rødseth, On a linear Diophantine problem of Frobenius, Journal für die reine und angewandte Mathematik 301 (1978) 171-178. DOI 10.1515/crll.1978.301.171, https://doi.org/10.1515/crll.1978.301.171
  15. A. Tripathi, Formulae for the Frobenius number in three variables, Journal of Number Theory 170 (2017) 368-389. DOI 10.1016/j.jnt.2016.05.027, https://doi.org/10.1016/j.jnt.2016.05.027
  16. J. Herzog, Generators and relations of abelian semigroups and semigroup rings, Manuscripta Mathematica 3(2) (1970) 175-193. DOI 10.1007/BF01273309, https://doi.org/10.1007/BF01273309
  17. O. N. Karpenkov, Geometry of Continued Fractions, Algorithms and Computation in Mathematics 26, Springer, 2013; 2nd edition 2022. DOI 10.1007/978-3-642-39368-6 (1st ed.), DOI 10.1007/978-3-662-65277-0 (2nd ed.), https://doi.org/10.1007/978-3-642-39368-6
  18. J. L. Ramírez Alfonsín, The Diophantine Frobenius Problem, Oxford Lecture Series in Mathematics and its Applications 30, Oxford University Press, 2005. DOI 10.1093/acprof:oso/9780198568209.001.0001, https://doi.org/10.1093/acprof:oso/9780198568209.001.0001