1a.lean`

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6 declarations, 5 contain sorry, 0 declared as axioms

Tao's Optimization constant 1a / An autocorrelation constant related to Sidon sets

References:

Tao's optimization constant 1a
[M2010] Matolcsi, Máté, and Carlos Vinuesa. "Improved bounds on the supremum of autoconvolutions." Journal of mathematical analysis and applications 372.2 (2010): 439-447. arXiv:0907.1379
[Y2026] Yuksekgonul, Mert et al., "Learning to Discover at Test Time," 2026, arXiv:2601.16175
[T2026] Together AI, "EinsteinArena-new-SOTA: State-of-the-art results on open math problems," 2026, GitHub
[PBV2026] Piterbarg, Andrei, Jai Bajaj, and Derrick Vincent. "A multi-scale arcsine lower bound for the Sidon autocorrelation constant $C_{1a}$," 2026, GitHub

Imports (1)

FormalConjecturesUtil

def Constant1a.C1a Tao's Optimization constant 1a / An autocorrelation constant related to Sidon sets: The largest constant line 50 source
Source documentation
Tao's Optimization constant 1a / An autocorrelation constant related to Sidon sets: The largest constant $C$ such that $$\max_{-1/2 \le t \le 1/2} \int_{\mathbb{R}} f(t - x) f(x)\,dx \ge C \left(\int_{-1/4}^{1/4} f(x)\,dx\right)^2$$ for all nonnegative $f \colon \mathbb{R} \to \mathbb{R}$.
The inequality is only required for square-integrable $f$. This does not change the constant (see [M2010, §2]), and it guarantees that every autoconvolution integral exists and that the supremum over $t$ is finite. This number is related to the maximal size of Sidon sets in additive combinatorics.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real
Proof uses: SupSet.sSup, Set.ofPred, LE.le, OfNat.ofNat, MeasureTheory.MemLp, ENNReal, MeasureTheory.MeasureSpace.volume, HMul.hMul, HPow.hPow, Nat, intervalIntegral, HDiv.hDiv, Neg.neg, Exists, And, Membership.mem, Set, Set.Icc, Eq, MeasureTheory.integral and 1 more
50 instances, coercions and auxiliary constants left out.
```
noncomputable def C1a : ℝ :=
…
```
theorem Constant1a.c1a_lower_bound A lower bound, proven by Matolcsi-Vinuesa in [M2010] sorry line 57 source
Source documentation
A lower bound, proven by Matolcsi-Vinuesa in [M2010].

depends on sorry what this means
Statement uses: LE.le, Real, OfScientific.ofScientific, Bool.true, Constant1a.C1a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
7 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem c1a_lower_bound : 1.2748 ≤ C1a := by
…
```
Mentions: Constant1a.C1a
theorem Constant1a.c1a_upper_bound An upper bound, proven by Yuksekgonul et al sorry line 62 source
Source documentation
An upper bound, proven by Yuksekgonul et al. in [Y2026].

depends on sorry what this means
Statement uses: LE.le, Real, Constant1a.C1a, OfScientific.ofScientific, Bool.true
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
7 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem c1a_upper_bound : C1a ≤ 1.5029 := by
…
```
Mentions: Constant1a.C1a
theorem Constant1a.mem_Ico_c1a How can the best known upper bound $1.5028628587\ldots$ of [T2026] be improved? sorry line 70 source
Source documentation
How can the best known upper bound $1.5028628587\ldots$ of [T2026] be improved?
The step function published with [T2026] gives $1.50286285870\ldots$. The figure $1.502862$ in the table of the first reference is this value truncated, so it is not a known upper bound.

depends on sorry what this means
Statement uses: Membership.mem, Real, Set, Set.Ico, Constant1a.C1a, OfScientific.ofScientific, Bool.true, sorryAx, Bool.false
Proof uses: Lean.Name, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
9 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem mem_Ico_c1a : answer(sorry) ∈ Set.Ico C1a 1.5028628587 := by
…
```
Mentions: Constant1a.C1a
theorem Constant1a.mem_Ioc_c1a How can the best known lower bound $1.292$ of [PBV2026] be improved? sorry line 75 source
Source documentation
How can the best known lower bound $1.292$ of [PBV2026] be improved?

depends on sorry what this means
Statement uses: Membership.mem, Real, Set, Set.Ioc, OfScientific.ofScientific, Bool.true, Constant1a.C1a, sorryAx, Bool.false
Proof uses: Lean.Name, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
9 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem mem_Ioc_c1a : answer(sorry) ∈ Set.Ioc 1.292 C1a := by
…
```
Mentions: Constant1a.C1a
theorem Constant1a.c1a_eq What is the exact value of the constant? sorry line 80 source
Source documentation
What is the exact value of the constant?

depends on sorry what this means
Statement uses: Eq, Real, Constant1a.C1a, sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem c1a_eq : C1a = answer(sorry) := by
…
```
Mentions: Constant1a.C1a