Theorems · Theorem · number theory
schnirelmannDensity_mul_le_card_filter
∀ {A : Set ℕ} [inst : DecidablePred fun x => x ∈ A] {n : ℕ},
schnirelmannDensity A * ↑n ≤ ↑{a ∈ Finset.Ioc 0 n | a ∈ A}.cardFor any natural n, the Schnirelmann density multiplied by n is bounded by |A ∩ {1, ..., n}|.
Note this property fails for the natural density.
- Defined in
- Mathlib.Combinatorics.Schnirelmann
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 118 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- DecidablePred
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites17
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- Realstatement · cited by 25,697
- Finset.cardstatement and proof · cited by 2,327
- MulZeroClass.mul_zeroproof · cited by 2,091
- Nat.cast_zeroproof · cited by 1,870
- eq_or_neproof · cited by 1,117
- Finset.filterstatement · cited by 949
- zero_leproof · cited by 382
- Finset.Iocstatement and proof · cited by 301
- Nat.cast_pos'proof · cited by 219
- lt_of_le_of_ne'proof · cited by 149
- le_div_iff₀proof · cited by 75
Cited by1
Results whose statement or proof uses this declaration.
- add_eq_univ_of_one_le_schirelmannDensity_add_schnirelmannDensityproof · cited by 0