Uniform syndeticity in multiple recurrence

The main theorem of this paper establishes a uniform syndeticity result concerning the multiple recurrence of measure-preserving actions on probability spaces. More precisely, for any integers d,l≥ 1 and any ε> 0 , we prove the existence of δ>0 and K≥ 1 (dependent only on d , l , and ε ) such that the following holds: Consider a solvable group Γ of derived length l , a probability space (X, μ ) , and d pairwise commuting measure-preserving Γ -actions T₁, … , Td on (X, μ ) . Let E be a measurable set in X with μ (E) ≥ ε . Then, K many (left) translates of $$ {align*} \{γ∈Γ μ(T_1^{γ⁻¹}(E)∩ T_2^{γ⁻¹} ∘ T^{γ⁻¹}_1(E)∩ ⋯ ∩ T^{γ⁻¹}_d∘ T^{γ⁻¹}d-1∘ ⋯ ∘ T^{γ⁻¹}_1(E))≥ δ \} {align*} $$ cover $Γ $ . This result extends and refines uniformity results by Furstenberg and Katznelson. As a combinatorial application, we obtain the following uniformity result. For any integers $d,l≥ 1$ and any $ε> 0$ , there are $δ>0$ and $K≥ 1$ (dependent only on d , l , and $ε $ ) such that for all finite solvable groups G of derived length l and any subset $E⊂ G^d$ with $m⊗ d(E)≥ ε $ (where m is the uniform measure on G ), we have that K -many (left) translates of $$ {align*} \{g∈ G &m⊗ d(\{(a_1,…,a_n)∈ G^d \\ & (a_1,…,a_n),(ga_1,a_2,…,a_n),…,(ga_1,ga_2,…, ga_n)∈ E\})≥ δ \} {align*} $$ cover G . The proof of our main result is a consequence of an ultralimit version of Austin’s amenable ergodic Szeméredi theorem.