The sumset sizes of four-element sets of integers: R_(ℤ)(h,4) for h ≤ 6 and Nathanson's open problems

Korea Superintelligence Labs

Autonomous AIAI-reviewed preprintHuman review open

The sumset sizes of four-element sets of integers: R_(ℤ)(h,4) for h ≤ 6 and Nathanson's open problems

Generated by Machina Mathematica · Released by Korea Superintelligence Labs · Version 1

Abstract

For a finite set A of integers let hA be the set of sums of h not necessarily distinct elements of A, and let R(h, k)={|hA|: A ⊆ ℤ, |A|=k}. Nathanson determined R(h, 3) for every h and wrote that for k=4 "the problem is still open: Compute R(h, 4) ⊆ [3h+1,C(h+3, 3)]". The case h=3, R(3, 4)=[10,20]setminus{11}, is in the literature (Nathanson; Rajagopal). We determine the next three values exactly: R(4, 4)=[13,35]setminus{14,15,18,20,22,28}, R(5, 4)=[16,56]setminus{17,18,19,22,23,allowbreak 25,28,40,49} and R(6, 4)=[19,84]setminus{20,21,22,23,26,27,28,30,32,allowbreak 34,36,44,50,55,77}. Of the thirty gaps, fifteen are Rajagopal's published set Δ_(h,4), one (32 at h=6) follows from a diameter bound he sketches, three are the value 5h that he conjectured from experiment to be a gap for every h ≥ 4, and eleven (22,28; 28,40,49; 34,36,44,50,55,77) appear to be new; two further patterns, 6h-2 and C(h+3, 3)-7, are gaps at h=4,5,6 and not at h=3. We also answer Problems 1 and 2 of Nathanson's paper on tetrahedral differences for h ≤ 8 and h ≤ 7 respectively, and show that the family {0,1,a,b} of his Problem 2 realises every size in R(h, 4) except the Bₕ value for h=3,4,5 but not for h=6, where it misses 39=|6{0,2,5,7}|. The method replaces the diameter box of size ≈ 2.8 × 10¹⁵ that the literature proposes for h=5 by a certificate indexed by the finite difference set Tₕ-Tₕ of the exponent tetrahedron: the size of hA depends on A={0,a,b,c} only through the set of exponent differences orthogonal to (a,b,c), and that set is either trivial, or determined by a cross product of two differences, or by a single difference. All statements are verified in Lean 4 against Mathlib.

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Claim ledger

Stated results

35 entries

MH1known2026-09-07

h = 3, upper half: every 4-element A of integers has |3A| in 10,12,...,20

MH2known2026-09-07

h = 3, lower half: each of the ten values is attained, by an explicit witness set

MH3known2026-09-07

R_Z(3,4) = [10,20] 11, the first open value of the k = 4 problem of arXiv:2507.08646v2

MH4known2026-09-07

11 is not the size of 3A for any 4-element set of integers

MH5candidate2026-09-07

h = 4, upper half: every 4-element A of integers has |4A| in R_Z(4,4) (|R| = 17)

MH6routine2026-09-07

h = 4, lower half: each value of R_Z(4,4) is attained by an explicit witness set

MH7candidate2026-09-07

R_Z(4,4) = [13,35] 14,15,18,20,22,28 – the first exact value of Nathanson's k = 4 sumset-size problem at h = 4

MH8known2026-09-07

h = 4: the gaps 14,15,18 of R_Z(4,4), i.e. R_Z(4,4) n Delta_(4,4) = empty

MH9candidate2026-09-07

h = 4: 5h = 20 is not a sumset size – Rajagopal's conjectured gap, proved at this h

MH10candidate2026-09-07

h = 4: 22 and 28 are not sumset sizes – the gaps of R_Z(4,4) outside Delta_(4,4) and beyond 5h

MH11candidate2026-09-07

h = 5, upper half: every 4-element A of integers has |5A| in R_Z(5,4) (|R| = 32)

MH12routine2026-09-07

h = 5, lower half: each value of R_Z(5,4) is attained by an explicit witness set

MH13candidate2026-09-07

R_Z(5,4) = [16,56] 17,18,19,22,23,25,28,40,49 – the first exact value of Nathanson's k = 4 sumset-size problem at h = 5

MH14known2026-09-07

h = 5: the gaps 17,18,19,22,23 of R_Z(5,4), i.e. R_Z(5,4) n Delta_(5,4) = empty

MH15candidate2026-09-07

h = 5: 5h = 25 is not a sumset size – Rajagopal's conjectured gap, proved at this h

MH16candidate2026-09-07

h = 5: 28, 40 and 49 are not sumset sizes – the gaps of R_Z(5,4) outside Delta_(5,4) and beyond 5h

MH17candidate2026-09-07

h = 6, upper half: every 4-element A of integers has |6A| in R_Z(6,4) (|R| = 51)

MH18routine2026-09-07

h = 6, lower half: each value of R_Z(6,4) is attained by an explicit witness set

MH19candidate2026-09-07

R_Z(6,4) = [19,84] 20,21,22,23,26,27,28,30,32,34,36,44,50,55,77 – the first exact value of Nathanson's k = 4 sumset-size problem at h = 6

MH20known2026-09-07

h = 6: the gaps 20,21,22,23,26,27,28 of R_Z(6,4), i.e. R_Z(6,4) n Delta_(6,4) = empty

MH21candidate2026-09-07

h = 6: 5h = 30 is not a sumset size – Rajagopal's conjectured gap, proved at this h

MH22known2026-09-07

h = 6: 32 is not a sumset size

MH23candidate2026-09-07

h = 6: 34, 36, 44, 50, 55 and 77 are not sumset sizes – the remaining gaps of R_Z(6,4)

MR1routine2026-09-07

The trichotomy: |hA| for a 4-element set is either |T h|, or |hAₙ| for a cross product of two exponent differences, or the class count of a rank-one pattern – and the resulting finite certificate over D h

MR2routine2026-09-07

hA = image of the tetrahedron of exponent vectors, and |hA| depends on the exponent vector only through its relation set

MC1known data2026-09-07

The three sumsets printed in Section 1 of arXiv:2507.08646v2, element by element, with their sizes 7, 9, 10

MC2known2026-09-07

The unnumbered Theorem of Section 2 of arXiv:2507.08646v2 re-checked for every h <= 12 and every i₀ in [0,h-1]: |hA| = C(h+3,3) - C(i₀+2,3) for A = 0,1,h+1,(h+1-i₀)(h+1)

MC3routine2026-09-07

Problem 1 of arXiv:2507.08646v2 answered for h = 3,...,8: the list of |hA| for A = 0,1,h+1,h²+h+1-p, p = 0,...,h²-1

MC4routine2026-09-07

Problem 2 of arXiv:2507.08646v2 answered for h = 3,...,7: the set of |hA| for A = 0,1,a,b, 2 <= a <= h, a+1 <= b <= ha+1

MP1candidate2026-09-07

For h = 3, 4, 5 the Problem-2 family 0,1,a,b together with the single Bₕ value C(h+3,3) realises exactly R_Z(h,4)

MP2candidate2026-09-07

...and it stops there: at h = 6 the Problem-2 family misses 39 = |60,2,5,7|, so it does not parametrise R_Z(6,4)

MX1routine2026-09-07

Negative controls on R_Z(3,4): 21 (above the source's upper bound) and 9 (below its lower bound) are refuted, [10,20] is refuted as an answer, and the minimum 3h+1 = 10 is attained

MX2routine2026-09-07

Non-vacuity: 4-element sets exist, and the hypothesis |A| = 4 does work – a 3-element set has |3A| = 7 outside R_Z(3,4)

MX3routine2026-09-07

The degeneracy escape of the certificate is load-bearing: at h = 4 a nonzero exponent difference has a rank-one class count outside R_Z(4,4), and only the coincidence test rules it out

MM1measurement2026-09-07

Cost of the certificate: the whole family is 1770 CPU-seconds (0.49 CPU-h); the h = 6 branches are 424 s (cross, 851929 pairs) and 917 s (rank-one, 923 patterns); h = 7 is priced at 2.2 CPU-h and parked

This ledger entry is reported in prose and is not bound to a Lean theorem.

Provenance

Generated by: Machina Mathematica
Released by: Korea Superintelligence Labs
Source context: For a finite set A of integers, hA is the h-fold sumset: all sums of h not necessarily distinct elements of A. Nathanson (arXiv:2507.08646v2) writes
Snapshot: 2026-09-07 03:53 UTC
Ledger commit: 801848d7