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module
public import FormalConjecturesUtilErdős Problem 867
References:
[CoPh96] Coppersmith, Don and Phillips, Steven, On a question of Erdős on subsequence sums. SIAM J. Discrete Math. (1996), 173-177.
[Fr93] Freud, R., Adding numbers - on a problem of P. Erdős. James Cook Mathematical Notes (1993), 6199-6202.
@[expose] public sectionopen Filternamespace Erdos867A finite set of naturals $A={a_1<\cdots<a_t}$ is consecutive-sum-free if it has no solutions to $a_i+a_{i+1}+\cdots+a_j\in A$ with $i<j$; equivalently, whenever an interval $[m,n]$ contains at least two elements of $A$, the sum of the elements of $A$ lying in $[m,n]$ is not itself an element of $A$.
def ConsecutiveSumFree (A : Finset ℕ) : Prop :=
∀ m n : ℕ, 2 ≤ (Finset.Icc m n ∩ A).card → (∑ a ∈ Finset.Icc m n ∩ A, a) ∉ AIs it true that if $A={a_1<\cdots <a_t}\subseteq {1,\ldots,N}$ has no solutions to $$a_i+a_{i+1}+\cdots+a_j\in A$$ then $$\lvert A\rvert \leq \frac{N}{2}+O(1)?$$
In fact this problem is false. Freud [Fr93] constructed a sequence with density $\geq 19/36$. The current best bounds are due to Coppersmith and Phillips [CoPh96], who prove that the maximal size of such an $A$ satisfies $$\frac{13}{24}N -O(1)\leq \lvert A\rvert \leq \left(\frac{2}{3}-\frac{1}{512}\right)N+\log N.$$
@[category research solved, AMS 5 11, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos867.lean"]
theorem erdos_867 : answer(False) ↔
∃ C : ℝ, ∀ N : ℕ, ∀ A ⊆ Finset.Icc 1 N, ConsecutiveSumFree A →
(A.card : ℝ) ≤ (N : ℝ) / 2 + C := ⊢ False ↔ ∃ C, ∀ (N : ℕ), ∀ A ⊆ Finset.Icc 1 N, ConsecutiveSumFree A → ↑A.card ≤ ↑N / 2 + C
All goals completed! 🐙Taking $A=(N/2,N]\cap \mathbb{N}$ shows $\lvert A\rvert \geq N/2-O(1)$ is possible.
N:ℕh2:N + 1 - (N / 2 + 1) = N - N / 2h3:N ≤ 2 * (N - N / 2)h4:↑N ≤ ↑(2 * (N - N / 2))⊢ ↑N / 2 ≤ ↑(N - N / 2)
N:ℕh2:N + 1 - (N / 2 + 1) = N - N / 2h3:N ≤ 2 * (N - N / 2)h4:↑N ≤ 2 * ↑(N - N / 2)⊢ ↑N / 2 ≤ ↑(N - N / 2)
All goals completed! 🐙Adenwalla has observed that $$\lvert A\rvert \leq (\tfrac{2}{3}+o(1))N.$$
@[category research solved, AMS 5 11]
theorem erdos_867.variants.adenwalla (ε : ℝ) (hε : 0 < ε) :
∀ᶠ N : ℕ in atTop, ∀ A ⊆ Finset.Icc 1 N, ConsecutiveSumFree A →
(A.card : ℝ) ≤ (2 / 3 + ε) * (N : ℝ) := ε:ℝhε:0 < ε⊢ ∀ᶠ (N : ℕ) in atTop, ∀ A ⊆ Finset.Icc 1 N, ConsecutiveSumFree A → ↑A.card ≤ (2 / 3 + ε) * ↑N
All goals completed! 🐙Freud [Fr93] constructed a sequence with density $\geq 19/36$.
@[category research solved, AMS 5 11]
theorem erdos_867.variants.freud :
∀ ε : ℝ, 0 < ε → ∀ᶠ N : ℕ in atTop, ∃ A ⊆ Finset.Icc 1 N, ConsecutiveSumFree A ∧
(((19 / 36 : ℝ) - ε) * (N : ℝ) ≤ (A.card : ℝ)) := ⊢ ∀ (ε : ℝ), 0 < ε → ∀ᶠ (N : ℕ) in atTop, ∃ A ⊆ Finset.Icc 1 N, ConsecutiveSumFree A ∧ (19 / 36 - ε) * ↑N ≤ ↑A.card
All goals completed! 🐙The current best bounds are due to Coppersmith and Phillips [CoPh96], who prove that the maximal size of such an $A$ satisfies $$\frac{13}{24}N -O(1)\leq \lvert A\rvert.$$
@[category research solved, AMS 5 11]
theorem erdos_867.variants.coppersmith_phillips_lower_bound :
∃ C : ℝ, ∀ N : ℕ, ∃ A ⊆ Finset.Icc 1 N, ConsecutiveSumFree A ∧
((13 / 24 : ℝ) * (N : ℝ) - C ≤ (A.card : ℝ)) := ⊢ ∃ C, ∀ (N : ℕ), ∃ A ⊆ Finset.Icc 1 N, ConsecutiveSumFree A ∧ 13 / 24 * ↑N - C ≤ ↑A.card
All goals completed! 🐙The current best bounds are due to Coppersmith and Phillips [CoPh96], who prove that the maximal size of such an $A$ satisfies $$\lvert A\rvert \leq \left(\frac{2}{3}-\frac{1}{512}\right)N+\log N.$$
@[category research solved, AMS 5 11]
theorem erdos_867.variants.coppersmith_phillips_upper_bound :
∀ᶠ N : ℕ in atTop, ∀ A ⊆ Finset.Icc 1 N, ConsecutiveSumFree A →
(A.card : ℝ) ≤ (2 / 3 - 1 / 512) * (N : ℝ) + Real.log (N : ℝ) := ⊢ ∀ᶠ (N : ℕ) in atTop, ∀ A ⊆ Finset.Icc 1 N, ConsecutiveSumFree A → ↑A.card ≤ (2 / 3 - 1 / 512) * ↑N + Real.log ↑N
All goals completed! 🐙end Erdos867