The true complexity of a polynomial progression in finite fields corresponds to the smallest-degree Gowers norm that controls the counting operator of the progression over finite fields of large characteristic. We give a conjecture that relates true complexity to algebraic relations between the terms of the progression, and we prove it for a number of progressions, including $x, x+y, x+y^{2}, x+y+y^{2}$ and $x, x+y, x+2y, x+y^{2}$ . As a corollary, we prove an asymptotic for the count of certain progressions of complexity 1 in subsets of finite fields. In the process, we obtain an equidistribution result for certain polynomial progressions, analogous to the counting lemma for systems of linear forms proved by Green and Tao.

We give necessary and sufficient conditions for joint ergodicity results of collections of sequences with respect to systems of commuting measure preserving transformations. Combining these results with a new technique that we call "seminorm smoothening", we settle several conjectures related to multiple ergodic averages of commuting transformations with polynomial iterates. We show that the Host-Kra factor is characteristic for pairwise independent polynomials, and that under certain ergodicity conditions the associated ergodic averages converge to the product of integrals. Moreover, when the polynomials are linearly independent, we show that the rational Kronecker factor is characteristic and deduce Khintchine-type lower bounds for the related multiple recurrence problem. Finally, we prove a nil plus null decomposition result for multiple correlation sequences of commuting transformations in the case where the iterates are given by families of pairwise independent polynomials. Introducingaveraged functions 30 5.5. Dual-difference interchange 31 5.6. Removing low complexity functions 33 6. Proof of Theorem 2.2 and 2.6 -Reduction to a degree lowering property 33 6.1. Preparation for induction 33 6.2. Property (P m−1 ) for the ergodic components 34 6.3. Replacing f m with an averaged function 36 6.4. Reduction to a degree lowering property 36 7. Proof of Theorems 2.2 and 2.6 -Proof of the degree lowering property 37 7.1. Ergodic decomposition and inverse theorem 37 7.2. Passing information to the ergodic components -Choice of (N ′ k ) 38 7.3. Dual-difference interchange and use of (P m−1 ) 38 7.4. Extracting a lower bound on ||| fm ||| s,Tm 40 8. Seminorm control and smoothening for pairwise independent polynomials 42 8.1. Seminorm smoothening for the family {n, n 2 , n 2 + n} 43 8.2. Introducing the formalism for longer families 46 8.3. The induction scheme 47 8.4. Proofs 51 Appendix A. Soft quantitative seminorm estimates 54 A.1. Main result 54 A.2. A more abstract statement 55 A.3. Mazur's lemma 57 A.4. Proof of Proposition A.2 57 A.5. Proof of the main result 60 A.6. Further extensions 61 Appendix B. PET bounds 61 References 644 For the implication (i) =⇒ (ii), instead of the last condition, we can assume good seminorm control for each system (X × X, µ × µ, Tj × Tj) for j ∈ [ℓ].5 Given a nilpotent Lie group G and a cocompact subgroup Γ, we call X = G/Γ a nilmanifold and

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Consider the sequence V(2, n) constructed in a greedy fashion by setting a 1 = 2, a 2 = n and defining a m+1 as the smallest integer larger than am that can be written as the sum of two (not necessarily distinct) earlier terms in exactly one way; the sequence V(2, 3), for example, is given by 3, 4, 5, 9, 10, 11, 16, 22, . . .We prove that if n 5 is odd, then the sequence V(2, n) has exactly two even terms {2, 2n} if and only if n − 1 is not a power of 2. We also show that in this case, V(2, n) eventually becomes a union of arithmetic progressions. If n − 1 is a power of 2, then there is at least one more even term 2n 2 + 2 and we conjecture there are no more even terms. In the proof, we display an interesting connection between V(2, n) and Sierpinski Triangle. We prove several other results, discuss a series of striking phenomena and pose many problems. This relates to existing results of Finch, Schmerl & Spiegel and a classical family of sequences defined by Ulam.2010 Mathematics Subject Classification. 11B13, 11B83.

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