feat(ErdosProblems/124): more statements

FormalConjectures/ErdosProblems/124.lean

@@ -15,11 +15,6 @@ limitations under the License.
 -/
 import FormalConjectures.Util.ProblemImports
 
-open Nat Finset Filter
-open scoped Real
-
-namespace Erdos124
-
 /-!
 # Erdős Problem 124
 
@@ -28,23 +23,81 @@ namespace Erdos124
 - [BEGL96] Burr, S. A. and Erdős, P. and Graham, R. L. and Li, W. Wen-Ching, Complete sequences of sets of integer powers. Acta Arith. (1996), 133-138.
 -/
 
+open Filter
+open scoped Pointwise
+
+namespace Erdos124
+
+/-- The set of integers which are the sum of distinct powers `d ^ i` with `i ≥ k`. -/
+def sumsOfDistinctPowers (d k : ℕ) : Set ℕ :=
+  {x | ∃ s : Finset ℕ, (∀ i ∈ s, k ≤ i) ∧ ∑ i ∈ s, d ^ i = x}
+
 /--
-Let $3\leq d_1 < d_2 < \cdots < d_k$ be integers such that
-$$\sum_{1\leq i\leq k}\frac{1}{d_i-1}\geq 1.$$
+Let  $3 \le d_1 < d_2 < \dots < d_r$ be integers such that
+$$\sum_{1 \le i \le r}\frac 1{d_i - 1} \ge 1.$$
 Can all sufficiently large integers be written as a sum of the shape $\sum_i c_ia_i$
-where $c_i\in \{0, 1\}$ and $a_i$ has only the digits $0, 1$ when written in base $d_i$?
+where $c_i \in \{0, 1\}$ and $a_i$ has only the digits $0, 1$ when written in base $d_i$?
+
+Conjectured by Erdős [Er97], solved by Boris Alexeev using Aristotle.
+-/
+@[category research solved, AMS 11]
+lemma erdos124.zero : answer(True) ↔
+    ∀ D : Finset ℕ, (∀ d ∈ D, 3 ≤ d) → 1 ≤ ∑ d ∈ D, (d - 1 : ℚ)⁻¹ →
+      ∀ᶠ n in atTop, n ∈ ∑ d ∈ D, sumsOfDistinctPowers d 0 := sorry
+
+/--
+Let $k \ne 0$ and $3\leq d_1 < d_2 < \cdots < d_r$ be integers of gcd equal to $1$ such that
+$$\sum_{1 \le i \le r}\frac 1{d_i - 1} \ge 1.$$
+Can all sufficiently large integers be written as a sum of the shape $\sum_i c_ia_i$
+where $c_i \in \{0, 1\}$ and $a_i$ is divisible by $d_i ^ k$ and has only the digits $0, 1$ when
+written in base $d_i$?
 
 Conjectured by Burr, Erdős, Graham, and Li [BEGL96]
 -/
 @[category research open, AMS 11]
-theorem erdos_124 : (∀ k, ∀ d : Fin k → ℕ,
-    (∀ i, 3 ≤ d i) →  StrictMono d → 1 ≤ ∑ i : Fin k, (1 : ℚ) / (d i - 1) →
-    ∀ᶠ n in atTop, ∃ c : Fin k → ℕ, ∃ a : Fin k → ℕ,
-    ∀ i, c i ∈ ({0, 1} : Finset ℕ) ∧
-    ∀ i, ((d i).digits (a i)).toFinset ⊆ {0, 1} ∧
-    n = ∑ i, c i * a i) ↔ answer(sorry) := by
+lemma erdos124.ne_zero : answer(sorry) ↔
+    ∀ k ≠ 0, ∀ D : Finset ℕ, (∀ d ∈ D, 3 ≤ d) → 1 ≤ ∑ d ∈ D, (d - 1 : ℚ)⁻¹ → D.gcd id = 1 →
+      ∀ᶠ n in atTop, n ∈ ∑ d ∈ D, sumsOfDistinctPowers d k :=
+  sorry
+
+/--
+All sufficiently large integers can be written as $a + b + c$ where $a$ has only the digits $0, 1$
+in base $3$, $b$ only the digits $0, 1$ in base $4$, $c$ only the digits $0, 1$ in base $7$.
+
+Provee by Burr, Erdős, Graham, and Li [BEGL96]
+-/
+@[category research solved, AMS 11]
+lemma erdos124.ne_zero_three_four_seven {k : ℕ} (hk : k ≠ 0) :
+    ∀ᶠ n in atTop,
+      n ∈ sumsOfDistinctPowers 3 k + sumsOfDistinctPowers 4 k + sumsOfDistinctPowers 7 k :=
   sorry
 
--- TODO(firsching): formalize the other two claims from the additional material
+/--
+Let $3\leq d_1 < d_2 < \cdots < d_r$ be integers such that all sufficiently large integers can be
+written as a sum of the shape $\sum_i c_ia_i$ where $c_i \in \{0, 1\}$ and $a_i$ has only the digits
+$0, 1$ when written in base $d_i$. Then
+$$\sum_{1 \le i \le r}\frac 1{d_i - 1} \ge 1.$$
+
+Reported by Burr, Erdős, Graham, and Li [BEGL96] as an observation of Pomerance
+-/
+@[category research solved, AMS 11]
+lemma erdos124.converse {D : Finset ℕ} (hD₃ : ∀ d ∈ D, 3 ≤ d)
+    (h : ∀ᶠ n in atTop, n ∈ ∑ d ∈ D, sumsOfDistinctPowers d 0) : 1 ≤ ∑ d ∈ D, (d - 1 : ℚ)⁻¹ :=
+  sorry
+
+/--
+For any $\varepsilon > 0$, there exists an infinite sequence $2 \le d_0 < d_1 < \dots$ such
+that all sufficiently large integer can be written as $\sum_{i \in I} a_i$ where $a_i$ has only
+the digits $0, 1$ when written in base $d_i$,
+but $\sum_{i \in I} \frac 1{d_i - 1} \le \varepsilon$.
+
+Proved by Melfi [Me04]
+-/
+@[category research solved, AMS 11]
+lemma erdos124.melfi_construction {ε : ℝ} (hε : 0 < ε) :
+    ∃ d : ℕ → ℕ, StrictMono d ∧ ∑' i, (d i - 1 : ℝ)⁻¹ ≤ ε ∧ ∀ᶠ n in atTop,
+      ∃ (I : Finset ℕ) (a : ℕ → ℕ), (∀ i ∈ I, a i ∈ sumsOfDistinctPowers (d i) 0) ∧
+        ∑ i ∈ I, a i = n :=
+  sorry
 
 end Erdos124