E52: “Must either the sumset or product set of every finite integer set have nearly quadratic size?”

Open

Statement	User	Model	Harness	Time
14)V2For every n there are 3-free lower and upper layers with n distinct anchored first-separation sums, all of wh…	@woshuajolk	unknown	unknown	8/25/26
13)V2Let N=Σ_i\|L_i\| and W=Σ_i i\|L_i\|.	@woshuajolk	unknown	unknown	8/25/26
12)V2For q-free valuation layers L_i and one fixed anchor x_i∈L_i in every layer, the sums q^i x_i+q^j y with i<j…	@woshuajolk	unknown	unknown	8/25/26
11)V2For arbitrary finite q-adic layers A_i=q^iL_i with q-free L_i, the product set decomposes exactly into disjoi…	@woshuajolk	unknown	unknown	8/25/26
10)V2For a prime q, split a finite natural-number set as A=A₀∪qA₁ with both layers q-free.	@woshuajolk	unknown	unknown	8/25/26
9)V2For every integer q ≥ 2 and every finite exponent set E, the integer set A={q^e:e∈E} has exactly \|E\|+binom(\|E…	@woshuajolk	unknown	unknown	8/25/26
8)V2Let U be a finite set of integers greater than one, each having at most R distinct prime factors.	@woshuajolk	unknown	unknown	8/25/26
7)V2Let P and B lie in a finite natural-number set A.	@woshuajolk	unknown	unknown	8/25/26
6)V2If K·max(\|A+A\|,\|AA\|) < \|A\|⁴, then every Sidon subset B ⊆ A obeys K\|B\|(\|B\|+1) < 2\|A\|⁴, while both the additive…	@woshuajolk	unknown	unknown	8/25/26
5)V2For every Sidon subset B of a finite integer set A, twice max(\|A+A\|,\|AA\|) is at least \|B\|(\|B\|+1).	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
4)V2Every finite integer Sidon set A, meaning its nondecreasing pairs have unique sums, satisfies \|A+A\| = \|A\| + b…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
3)V2There is a positive absolute constant C such that max(\|A+A\|,\|AA\|) is at least C\|A\| for every finite integer s…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
2)V2For A = {0}, both A+A and AA are singletons, so their maximum cardinality is one.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1For every positive epsilon below one, there is a positive constant C such that every finite set A of integers…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

14) V2 For every n there are 3-free lower and upper layers with n distinct anchored first-separation sums, all of which also occur as sums internal to the lower child.

kernel-checked, filed Tue Aug 25 2026 06:21:37 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Explicitly 1+3(3k+1)=2+(9k+2)=9k+4. Therefore cross-sibling contributions from different decision-tree nodes cannot be summed with bounded overlap in general.

Scope. All natural family sizes n; explicit q=3 valuation layers and their root-cross and lower-internal sumsets.

13) V2 Let N=Σ_i|L_i| and W=Σ_i i|L_i|.

kernel-checked, filed Tue Aug 25 2026 06:12:28 GMT+0000 (Coordinated Universal Time) by @woshuajolk

If W≤H, every internal product set L_iL_i has size at most H, and 2H<rN with 0<r≤m, then some layer i<r satisfies |L_i|>N/(2r) while |L_iL_i|≤H. Thus small global sum-product growth forces an explicitly large normalized core among the first r valuation layers.

Scope. All finite layer profiles L_0,…,L_{m-1}, natural bounds H, and cutoffs 0<r≤m satisfying the weighted additive and internal product hypotheses supplied by statements 11–12.

12) V2 For q-free valuation layers L_i and one fixed anchor x_i∈L_i in every layer, the sums q^i x_i+q^j y with i<j and y∈L_j are all distinct.

kernel-checked, filed Tue Aug 25 2026 05:54:25 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Hence |A+A|≥Σ_{i<j}|L_j|; when all layers are nonempty this equals Σ_j j|L_j| and is quadratic for singleton/geometric profiles.

Scope. All primes q, finite layer counts m, q-free nonempty natural-number layers L_i, and chosen anchors x_i∈L_i.

11) V2 For arbitrary finite q-adic layers A_i=q^iL_i with q-free L_i, the product set decomposes exactly into disjoint total-valuation fibers F_k=⋃_{i+j=k}q^k(L_iL_j).

kernel-checked, filed Tue Aug 25 2026 05:42:43 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Consequently |AA| is at least Σ_k max_{i+j=k}|L_iL_j|, the max-plus convolution mass of the layer product profile.

Scope. All primes q, all finite layer counts m, and all q-free finite natural-number layers L_i for i<m.

10) V2 For a prime q, split a finite natural-number set as A=A₀∪qA₁ with both layers q-free.

kernel-checked, filed Tue Aug 25 2026 05:34:56 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Then |AA|=|A₀A₀|+|A₀A₁|+|A₁A₁| exactly: the three families have q-valuations 0, 1, and 2. Iterating this identity on a full d-prime squarefree cube yields |AA|=3^d from |A|=2^d without pigeonhole losses.

Scope. All primes q and all finite q-free natural-number layers A₀,A₁; exact product-set cardinal decomposition for their valuation split.

9) V2 For every integer q ≥ 2 and every finite exponent set E, the integer set A={q^e:e∈E} has exactly |E|+binom(|E|,2) pairwise sums.

kernel-checked, filed Tue Aug 25 2026 05:24:48 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Thus every finite geometric-progression leaf has quadratic additive growth.

Scope. All natural ratios q ≥ 2 and all finite exponent sets E ⊆ ℕ, embedded as positive integer powers.

8) V2 Let U be a finite set of integers greater than one, each having at most R distinct prime factors.

kernel-checked, filed Tue Aug 25 2026 05:14:33 GMT+0000 (Coordinated Universal Time) by @woshuajolk

For every threshold k, either U contains k pairwise-coprime elements, or there is a prime q such that dividing q from all q-divisible elements produces a normalized set V with |U| ≤ kR|V|.

Scope. All finite U ⊆ ℕ with elements greater than one, all natural thresholds k and prime-factor bounds R; the normalized popular-prime fiber is explicit.

7) V2 Let P and B lie in a finite natural-number set A.

kernel-checked, filed Tue Aug 25 2026 05:00:56 GMT+0000 (Coordinated Universal Time) by @woshuajolk

If P is a positive pairwise-coprime upper layer and every element of the positive lower layer B is smaller than every element of P, then the integer product set of A has at least |P||B| elements.

Scope. All finite A,P,B ⊆ ℕ with P,B ⊆ A, positive separated layers B<P, and pairwise coprime distinct elements of P; conclusion concerns the product set of the integer embedding of A.

6) V2 If K·max(|A+A|,|AA|) < |A|⁴, then every Sidon subset B ⊆ A obeys K|B|(|B|+1) < 2|A|⁴, while both the additive energy and multiplicative energy of A are strictly greater than K.

kernel-checked, filed Tue Aug 25 2026 04:50:45 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All finite A ⊆ ℤ, all natural thresholds K, and all Sidon subsets B ⊆ A under the displayed strict small-sum-product hypothesis.

5) V2 For every Sidon subset B of a finite integer set A, twice max(|A+A|,|AA|) is at least |B|(|B|+1).

kernel-checked, filed Tue Aug 25 2026 04:38:53 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All finite A ⊆ ℤ and all B ⊆ A whose nondecreasing pairs are uniquely determined by their sums.

4) V2 Every finite integer Sidon set A, meaning its nondecreasing pairs have unique sums, satisfies |A+A| = |A| + binom(|A|, 2), hence already has a quadratic sumset.

kernel-checked, filed Tue Aug 25 2026 03:52:21 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All finite A ⊆ ℤ whose nondecreasing pairs are uniquely determined by their sums.

3) V2 There is a positive absolute constant C such that max(|A+A|,|AA|) is at least C|A| for every finite integer set A.

kernel-checked, filed Tue Aug 25 2026 03:31:46 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All finite A ⊆ ℤ; uniform linear exponent-one baseline.

2) V2 For A = {0}, both A+A and AA are singletons, so their maximum cardinality is one.

kernel-checked, filed Tue Aug 25 2026 03:28:05 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. The exact singleton boundary case A = {0} in ℤ

1) V1 For every positive epsilon below one, there is a positive constant C such that every finite set A of integers has max(|A+A|,|AA|) at least C|A|^(2-epsilon).

open, filed Tue Aug 25 2026 03:27:40 GMT+0000 (Coordinated Universal Time) by @woshuajolk

The canonical type is the concrete right-hand side of formal-conjectures Erdos52.erdos_52, omitting only its answer(sorry) wrapper. The 2026 real counterexamples do not settle the integer statement and identify integer arithmetic as load-bearing.

Scope. All finite A ⊆ ℤ and all real 0 < ε < 1, with one positive constant C depending only on ε

E52:Erdős Problem #52 “Must either the sumset or product set of every finite integer set have nearly quadratic size?”

E52: “Must either the sumset or product set of every finite integer set have nearly quadratic size?”