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We obtain new results pertaining to convergence and recurrence of multiple ergodic averages along functions from a Hardy field. Among other things, we confirm some of the conjectures posed by Frantzikinakis in [19,21] and obtain combinatorial applications which contain, as rather special cases, several previously known (polynomial and non-polynomial) extensions of Szemerédi's theorem on arithmetic progressions [7,8,10,19,24]. One of the novel features of our results, which is not present in previous work, is that they allow for a mixture of polynomials and non-polynomial functions. As an illustration, assume f i (t) = a i , 1 t c i , 1 + ⋯ + a i , d t c i , d for c i , j > 0 and a i , j ∈ R. Then • for any measure preserving system (X , B , μ , T) and h 1 , ... , h k ∈ L ∞ (X) , the limit lim N → ∞ ⁡ 1 N ∑ n = 1 N T [ f 1 (n) ] h 1 ⋯ T [ f k (n) ] h k exists in L 2 ; • for any E ⊂ N with d ‾ (E) > 0 there are a , n ∈ N such that { a , a + [ f 1 (n) ] , ... , a + [ f k (n) ] } ⊂ E. We also show that if f 1 , ... , f k belong to a Hardy field, have polynomial growth, and are such that no linear combination of them is a polynomial, then for any measure preserving system (X , B , μ , T) and any A ∈ B , lim sup N → ∞ 1 N ∑ n = 1 N μ (A ∩ T − [ f 1 (n) ] A ∩ ... ∩ T − [ f k (n) ] A) ⩾ μ (A) k + 1. [ABSTRACT FROM AUTHOR]

In 1977 L.T. Ramsey showed that any sequence in Z 2 with bounded gaps contains arbitrarily many collinear points. Thereafter, in 1980, C. Pomerance provided a density version of this result, relaxing the condition on the sequence from having bounded gaps to having gaps bounded on average. We give a higher dimensional generalization of these results. Our main theorem is the following. Theorem . Let d ∈ N , let f : Z d → Z d + 1 be a Lipschitz map and let A ⊂ Z d have positive upper Banach density. Then f ( A ) contains arbitrarily many collinear points. Note that Pomerance’s theorem corresponds to the special case d = 1 . In our proof, we transfer the problem from a discrete to a continuous setting, allowing us to take advantage of analytic and measure theoretic tools such as Rademacher’s theorem. [ABSTRACT FROM AUTHOR]

We establish new recurrence and multiple recurrence results for a rather large family F of non-polynomial functions which contains tempered functions and (non-polynomial) functions from a Hardy field with polynomial growth. In particular, we show that, somewhat surprisingly (and in the contrast to the multiple recurrence along polynomials), the sets of return times along functions from F are thick , i.e., contain arbitrarily long intervals. A major component of our paper is a new result about equidistribution of sparse sequences on nilmanifolds, whose proof borrows ideas from the work of Green and Tao [26]. Among other things, we show that for any f ∈ F , any invertible probability measure preserving system (X , B , μ , T) , any A ∈ B with μ (A) > 0 , and any ε > 0 , the sets of returns { n ∈ N : μ (A ∩ T − ⌊ f (n) ⌋ A) > μ 2 (A) − ε } { n ∈ N : μ (A ∩ T − ⌊ f (n) ⌋ A ∩ T − ⌊ f (n + 1) ⌋ A ∩ ⋯ ∩ T − ⌊ f (n + k) ⌋ A) > 0 } are thick. Our recurrence theorems imply, via Furstenberg's correspondence principle, some new combinatorial results. For example, we show that given a set E ⊂ N with positive upper density, for every k ∈ N there are a , n ∈ N such that { a , a + ⌊ f (n) ⌋ , ⋯ , a + ⌊ f (n + k) ⌋ } ⊂ E. When f (n) = n c , with c > 0 non-integer, this result provides a positive answer to a question posed by Frantzikinakis [16, Problem 23]. [ABSTRACT FROM AUTHOR]

In the 1970s Deuber introduced the notion of ( m , p , c ) -sets in N and showed that these sets are partition regular and contain all linear partition regular configurations in N . In this paper we obtain enhancements and extensions of classical results on ( m , p , c ) -sets in two directions. First, we show, with the help of ultrafilter techniques, that Deuber's results extend to polynomial configurations in abelian groups. In particular, we obtain new partition regular polynomial configurations in Z d . Second, we give two proofs of a generalization of Deuber's results to general commutative semigroups. We also obtain a polynomial version of the central sets theorem of Furstenberg, extend the theory of ( m , p , c ) -systems of Deuber, Hindman and Lefmann and generalize a classical theorem of Rado regarding partition regularity of linear systems of equations over N to commutative semigroups. [ABSTRACT FROM AUTHOR]

Building on previous work of Di Nasso and Luperi Baglini, we provide general necessary conditions for a Diophantine equation to be partition regular. These conditions are inspired by Rado's characterization of partition regular linear homogeneous equations. We conjecture that these conditions are also sufficient for partition regularity, at least for equations whose corresponding monovariate polynomial is linear. This would provide a natural generalization of Rado's theorem. We verify that such a conjecture holds for the equations x 2 − x y + a x + b y + c z = 0 and x 2 − y 2 + a x + b y + c z = 0 for a , b , c ∈ Z such that a b c = 0 or a + b + c = 0. To deal with these equations, we establish new results concerning the partition regularity of polynomial configurations in Z such as x , x + y , x y + x + y , building on the recent result on the partition regularity of x , x + y , x y. [ABSTRACT FROM AUTHOR]