Multiple ergodic averages along functions from a Hardy field: Convergence, recurrence and combinatorial applications

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 Szemer & eacute;di'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) + <middle dot> <middle dot> <middle dot> + a (i,d) t( c i,d) for c (i,j )> 0 and a (i,j) is an element of R. Then center dot for any measure preserving system (X, B, mu, T) and h( 1) , ... , h (k) E L- infinity (X), the limit lim (N ->infinity) 1/ N Sigma(N) T-n=1 ([f 1 (n)] )h (1) <middle dot> <middle dot> <middle dot> T ([f k (n )]) h( k )exists in L (2) ; center dot for any E subset of N with d(E) > 0 there are a, n is an element of N such that {a, a + [f (1) (n)], ... , a + [f (k) (n )]} subset of 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, mu, T ) and any A is an element of B , lim sup (N ->infinity) 1 /N Sigma(N) (n =1) mu( A boolean AND T( - [ f 1 ( n )] )A boolean AND...boolean AND T (- [ f k ( n )]) A ) >= mu ( A ) (k +1) . (c) 2024 Published by Elsevier Inc.