Combinatorics in groups

Additive and Ramsey-type combinatorics inside groups: approximate subgroups and product theorems, small doubling, partitions and colourings of groups, dense and coset-free partitions, subset factorizations, products of large normal subsets.

• Approximate subgroups
• Coset-free partitions of groups
• Dense partitions of groups
• Direct products of subsets of groups
• Disjoint coset systems
• Factorizations of groups into subsets
• Monochromatic symmetric subsets of abelian groups
• Nilpotent pairs in finite subsets of groups
• Partitions meeting every infinite coset
• Partitions of groups of singular cardinality
• Product growth in special linear groups over finite fields
• Products of large normal subsets
• Products of permuted sequences in groups
• Separated subsets of groups

TheoremProduct theorem for special linear groups#●ready to prove

Let \(n\) be a natural number. For every real \(\varepsilon\) with \(0 < \varepsilon < 1\) there is a real \(\delta > 0\) such that the following holds for every finite field \(k\): if \(A\) is a subset of \(\mathrm{SL}(n, k)\) with \(A^{-1} = A\) and \(1 \in A\) that generates \(\mathrm{SL}(n, k)\), then \(|A|^{1+\delta} \le |A^3|\) or \(|\mathrm{SL}(n, k)|^{1-\varepsilon} \le |A|\), where \(A^3 = \{abc : a, b, c \in A\}\). Thus \(\delta\) depends only on \(n\) and \(\varepsilon\).

theorem_wanted product_theorem_special_linear {n : ℕ} :
    ∀ ε : ℝ, 0 < ε → ε < 1 →
      ∃ δ : ℝ, 0 < δ ∧
        ∀ (k : Type*) [Field k] [Fintype k] [DecidableEq k]
          (A : Finset (Matrix.SpecialLinearGroup (Fin n) k))
          (_hsymm : A⁻¹ = A)
          (_hone : 1 ∈ A)
          (_hgen : Subgroup.closure (A : Set (Matrix.SpecialLinearGroup (Fin n) k)) = ⊤),
          Real.rpow (Finset.card A : ℝ) (1 + δ)
            ≤ (Finset.card (A ^ 3) : ℝ) ∨
            Real.rpow (Fintype.card (Matrix.SpecialLinearGroup (Fin n) k) : ℝ) (1 - ε)
              ≤ (Finset.card A : ℝ)

Sources

• S. Machado, Topics in Geometric Group Theory lecture notes, https://sites.google.com/view/simonmachado/teaching, Theorem 1.9; originals B. Breuillard, B. Green, T. Tao, Approximate subgroups of linear groups, Geom. Funct. Anal. 21 (2011), and L. Pyber, E. Szabó, Growth in finite simple groups of Lie type, J. Amer. Math. Soc. 29 (2016).

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