1097.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

4 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 1097

References:

erdosproblems.com/1097
[Bo99] Bourgain, J., On the dimension of {K}akeya sets and related maximal inequalities. Geom. Funct. Anal. (1999), 256--282
[KaTa99] Katz, Nets Hawk and Tao, Terence, Bounds on arithmetic projections, and applications to the {K}akeya conjecture. Math. Res. Lett. (1999), 625--630.
[Le15] Lemm, Marius, New counterexamples for sums-differences. Proc. Amer. Math. Soc. (2015), 3863--3868.
[GGTW25] B. Georgiev, J. Gómez-Serrano, T. Tao, and A. Wagner, Mathematical exploration and discovery at scale. arXiv:2511.02864 (2025).

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100OpenSet1.lean

def Erdos1097.CommonDifferencesThreeTermAP Documented declaration: CommonDifferencesThreeTermAP line 43 source
Source documentation
Given a finite set of integers A (modelled as a Finset ℤ), the set CommonDifferencesThreeTermAP A consists of all integers d such that there is a non-trivial three-term arithmetic progression a, b, c ∈ A with b - a = d and c - b = d.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, Int, Set
Proof uses: Set.ofPred, And, Ne, OfNat.ofNat, Exists, Membership.mem, Eq, HSub.hSub
5 instances, coercions and auxiliary constants left out.
```
def CommonDifferencesThreeTermAP (A : Finset ℤ) : Set ℤ :=
…
```
theorem Erdos1097.erdos_1097 sorry line 65 source
depends on sorry what this means
Statement uses: Iff, False, Exists, Real, And, GT.gt, OfNat.ofNat, Finset, Int, LE.le, Set.ncard, Erdos1097.CommonDifferencesThreeTermAP, HMul.hMul, HPow.hPow, Finset.card, HDiv.hDiv, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos1097.erdos_1097.variants.weaker A weaker bound has been proven: there are always at most $n^2$ such values of $d$ line 73 source
Source documentation
A weaker bound has been proven: there are always at most $n^2$ such values of $d$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, Int, LE.le, Nat, Set.ncard, Erdos1097.CommonDifferencesThreeTermAP, HPow.hPow, Finset.card, OfNat.ofNat
Proof uses: Finset.image, Prod, HSub.hSub, Prod.snd, Prod.fst, SProd.sprod, Set, Membership.mem, Ne, Exists, And, Eq, Iff.mpr, Finset.mem_coe, Finset.mem_image, Exists.intro, Prod.mk, And.intro, Finset.mem_product, Trans.trans and 8 more
31 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1097.variants.weaker :
    ∀ A, (CommonDifferencesThreeTermAP A).ncard ≤ A.card ^ 2 := by
…
```
Mentions: Erdos1097.CommonDifferencesThreeTermAP
theorem Erdos1097.erdos_1097.variants.lower_bound A trivial lower bound: for sufficiently large n there exist sets line 93 source
Source documentation
A trivial lower bound: for sufficiently large n there exist sets $A$ with $|A| = n$ that contain at least $\Omega(n)$ distinct common differences of three-term arithmetic progressions.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Filter.Eventually, Nat, Finset, Int, Eq, Finset.card, LE.le, HMul.hMul, Set.ncard, Erdos1097.CommonDifferencesThreeTermAP, Filter.atTop
Proof uses: Exists.intro, HDiv.hDiv, And.intro, Mathlib.Meta.NormNum.isNNRat_lt_true, Mathlib.Meta.NormNum.IsNat.to_isNNRat, Mathlib.Meta.NormNum.isNat_ofNat, Nat.cast_zero, Mathlib.Meta.NormNum.isNNRat_div, Mathlib.Meta.NormNum.isNNRat_mul, Inv.inv, Eq.refl, Nat.cast_one, Mathlib.Meta.NormNum.isNNRat_inv_pos, Mathlib.Meta.NormNum.instAtLeastTwo, Nat.mul, Bool, Bool.true, Eq.mpr, id, congrArg and 114 more
162 instances, coercions and auxiliary constants left out.

Tactic trace: 56 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1097.variants.lower_bound : ∃ c > (0 : ℝ), ∀ᶠ n in Filter.atTop, ∃ (A : Finset ℤ),
    A.card = n ∧ c * (n : ℝ) ≤ (CommonDifferencesThreeTermAP A).ncard := by
…
```
Mentions: c, Erdos1097.CommonDifferencesThreeTermAP