We set ? = n M k . Let (n, n ) ? A × B with max(n, n ) ? ? ,

, Theorem 5.15 holds when we restrict ourselves to p = q

, There exists B ? A with positive lower density, min(B) ? a and |n ? n | ? a for all n, Lemma 5.17. Let A ? N with positive lower density and a > 0

, A = {n j : j ? N} in an increasing order and define B = {n ka : k ? N}. We then go inductively from two sets to a sequence of sets

, There exists a sequence (A(p)) of pairwise disjoint subsets of N such that (i) each set A(p) has positive lower density

, ii) for all C > 0, there exists ? > 0 such that, for all (n, n ) ? A(p) × A(q) with p = q and max(n, n ) ? ?

, We shall construct by induction two sequences of sets (A(p)) and (B(p)) and a sequence of integers (? k ) such that, at each step r, (a) for all 1 ? p ? r, A(p) and B(p) are disjoint and have positive lower density

A(q) ? B(p) and B(q) ? B(p) ,

, A(p) and n ? B(q), max(n, n ) ? ? =? |n ? n | ? C

Condition (c) is only helpful for the induction hypothesis. We initialize the construction by applying Lemma 5.16 to E = N. We set A(1) = A and B(1) = B which satisfy (a), (b) and (c). In particular, applying (c) now that the construction has been done until step r and let us perform it for step r + 1. Let E be a subset of B(r) with positive lower density and |n ? n | ? r + 1 provided n = n are in E. We apply Lemma 5.16 to this set E and we set A(r + 1) = A and B(r + 1) = B, so that (a) and (b) are clearly satisfied, r}, for all 1 ? p ? q ? r, for all n ? A(p) and n ? B(q), vol.18 ,

, The proof of (d) is slightly more delicate. For k = 1, . . . , r, we have to verify that for 1 ? p ? r + 1, n ? A(p) and n ? B(r + 1), max(n, n ) ? ? k =? |n ? n | ? k

Hypercyclic and Cyclic Vectors, Journal of Functional Analysis, vol.128, issue.2, pp.374-383, 1995. ,

Existence of hypercyclic operators on topological vector spaces, J. Funct. Anal, vol.148, pp.384-390, 1997. ,

Seoane-Sepúlveda. Powers of hypercyclic functions for some classical hypercyclic operators. Integral Equations Operator Theory, vol.58, pp.591-596, 2007. ,

Hypercyclic algebras, J. Funct. Anal, vol.276, pp.3441-3467, 2019. ,

URL : https://hal.archives-ouvertes.fr/hal-01758833

Dynamics of linear operators, Cambridge Tracts in Math, vol.179, 2009. ,

URL : https://hal.archives-ouvertes.fr/hal-00410866

Difference sets and frequently hypercyclic weighted shifts Ergodic Theory Dynam, Systems, vol.35, pp.691-709, 2015. ,

On hypercyclic operators on Banach spaces, Proc. Amer. Math. Soc, vol.127, issue.4, pp.1003-1010, 1999. ,

Non-finite-dimensional closed vector spaces of universal functions for composition operators, J. Approx. Theory, vol.82, issue.3, pp.375-391, 1995. ,

Convolution operators supporting hypercyclic algebras, J. Math. Anal. Appl, vol.445, pp.1232-1238, 2017. ,

Hypercyclic algebras for convolution and composition operators, J. Funct. Anal, vol.274, pp.2884-2905, 2018. ,

Algebrable sets of hypercyclic vectors for convolution operators, Israel J. Math, 2020. ,

Hypercyclic Algebras for convolution operators of unimodular constant term ArXiv e-prints, 2019. ,

On a class of B 0 -spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys, vol.5, pp.379-383, 1957. ,

Upper frequent hypercyclicity and related notions, Rev. Mat. Complut, vol.31, pp.673-711, 2018. ,

Invariant manifolds of hypercyclic vectors, Proc. Amer. Math. Soc, vol.118, pp.845-847, 1993. ,

Algebrability of the set of hypercyclic vectors for backward shift operators, Adv. Math, vol.366, pp.25-47, 2020. ,

Algebras of frequently hypercyclic vectors, Math. Nachr, vol.293, issue.6, pp.1120-1135, 2020. ,

Linear chaos, 2011. ,

Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal, vol.98, pp.229-269, 1991. ,

Hypercyclic subspaces and weighted shifts, Adv. Math, vol.255, pp.305-337, 2014. ,

Hypercyclic subspaces for Fréchet space operators, J. Math. Anal. Appl, vol.319, pp.764-782, 2006. ,

On the set of hypercyclic vectors for the differentiation operator, Israel J. Math, vol.180, pp.271-283, 2010. ,