254.lean`

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4 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 254

References:

erdosproblems.com/254
[Ca60] Cassels, J. W. S., On the representation of integers as the sums of distinct summands taken from a fixed set. Acta Sci. Math. (Szeged) (1960), 111-124.

Imports (1)

FormalConjecturesUtil

def Erdos254.IsSumOfDistinct An integer n can be written as a sum of distinct elements of A line 38 source
Source documentation
An integer n can be written as a sum of distinct elements of A.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Nat
Proof uses: Exists, Finset, And, LE.le, Eq, Finset.sum
4 instances, coercions and auxiliary constants left out.
```
def IsSumOfDistinct (A : Set ℕ) (n : ℕ) : Prop :=
…
```
theorem Erdos254.not_summable_iff_tendsto_partial_sums Documented declaration: not_summable_iff_tendsto_partial_sums line 46 source
Source documentation
The hypothesis ¬ Summable (fun n : A ↦ distToNearestInt (θ * n)) used below says exactly that the partial sums of ‖θ n‖ over n ∈ A diverge, which is the form the linked proof uses. distToNearestInt is nonnegative, so this is an instance of not_summable_subtype_iff_tendsto_sum_indicator.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Nat, Real, Iff, Not, Summable, distToNearestInt, HMul.hMul, Membership.mem, SummationFilter.unconditional, Filter.Tendsto, Finset.sum, Finset.range, Set.indicator, Filter.atTop
Proof uses: not_summable_subtype_iff_tendsto_sum_indicator, distToNearestInt_nonneg
18 instances, coercions and auxiliary constants left out.
```
theorem not_summable_iff_tendsto_partial_sums (A : Set ℕ) (θ : ℝ) :
    ¬ Summable (fun n : A ↦ distToNearestInt (θ * (n : ℝ))) ↔
      Tendsto (fun N : ℕ =>
          ∑ n ∈ Finset.range N, A.indicator (fun n => distToNearestInt (θ * (n : ℝ))) n)
        atTop atTop :=
…
```
Mentions: distToNearestInt
theorem Erdos254.erdos_254 Documented declaration: erdos_254 sorry line 61 source
Source documentation
Let $A\subseteq \mathbb{N}$ be such that $\lvert A\cap [1,2x]\rvert -\lvert A\cap [1,x]\rvert \to \infty\textrm{ as }x\to \infty$ and $\sum_{n\in A} \{ \theta n\}=\infty$ for every $\theta\in (0,1)$, where $\{x\}$ is the distance of $x$ from the nearest integer. Then every sufficiently large integer is the sum of distinct elements of $A$.

depends on sorry what this means
Statement uses: Set, Nat, And, Filter.Tendsto, HSub.hSub, Set.ncard, Inter.inter, Set.Icc, OfNat.ofNat, HMul.hMul, Filter.atTop, Real, LT.lt, Not, Summable, distToNearestInt, Membership.mem, SummationFilter.unconditional, Filter.Eventually, Erdos254.IsSumOfDistinct
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
44 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_254 :
    ∀ (A : Set ℕ),
      (Tendsto (fun x : ℕ ↦ (A ∩ Icc 1 (2 * x)).ncard - (A ∩ Icc 1 x).ncard) atTop atTop) ∧
      (∀ θ : ℝ, 0 < θ → θ < 1 → ¬ Summable (fun n : A ↦ distToNearestInt (θ * (n : ℝ)))) →
        ∀ᶠ m in atTop, IsSumOfDistinct A m := by
…
```
Mentions: distToNearestInt, Erdos254.IsSumOfDistinct
theorem Erdos254.erdos_254.variants.cassels Cassels [Ca60] proved this under the alternative hypotheses sorry line 74 source
Source documentation
Cassels [Ca60] proved this under the alternative hypotheses $\lim \frac{\lvert A\cap [1,2x]\rvert -\lvert A\cap [1,x]\rvert}{\log\log x}=\infty$ and $\sum_{n\in A} \{ \theta n\}^2=\infty$ for every $\theta\in (0,1)$.

depends on sorry what this means
Statement uses: Set, Nat, And, Filter.Tendsto, Real, HDiv.hDiv, HSub.hSub, Set.ncard, Inter.inter, Set.Icc, OfNat.ofNat, HMul.hMul, Real.log, Filter.atTop, LT.lt, Not, Summable, HPow.hPow, distToNearestInt, Membership.mem and 3 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
60 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_254.variants.cassels :
    ∀ (A : Set ℕ),
      (Tendsto (fun x : ℕ ↦ (((A ∩ Icc 1 (2 * x)).ncard : ℝ) -
        ((A ∩ Icc 1 x).ncard : ℝ)) / Real.log (Real.log x)) atTop atTop) ∧
      (∀ θ : ℝ, 0 < θ → θ < 1 → ¬ Summable (fun n : A ↦ (distToNearestInt (θ * (n : ℝ)))^2)) →
        ∀ᶠ m in atTop, IsSumOfDistinct A m := by
…
```
Mentions: distToNearestInt, Erdos254.IsSumOfDistinct