Normed versus topological groups: Dichotomy and duality

The Steinhaus theorem and regular variation: de Bruijn and after

“…A comprehensive examination of results of this kind was recently given in [BinO11], from the point of view of in…nite combinatorics; see also [BinO6]. In particular, the subgroup dichotomy is relevant to the area of Ramsey theory [BinO8].…”

“…[Bal]). One can also work with normed groups [BinO6]. Here the dichotomy takes the form: normed groups are either topological or pathological.…”

Beurling regular variation, Bloom dichotomy, and the Gołąb–Schinzel functional equation

“…with ' = 1) is either self-neglecting or pathological -extends to W SE: when ' 2 W SE either ' 2 SE; or ' is 'pathological'. (For other occurrences of dichotomy in this area see [BinO3,4,5].) Indeed, ' 2 W SE says merely that the limit function ' is well-de…ned, but nothing about whether ' satis…es (GS).…”

Infinite combinatorics in function spaces: Category methods

“…We now specialize Theorem 1 to a metric group setting in order to consider sequences of autohomeomorphisms generated as shifts h n (x) = xz n : Let T be a normed group with norm jjtjj := d(t; e T ), where d is right-invariant (see [BOst12] for background and references). Thus d(x; y) = d(e; yx 1 ) = jjyx 1 jj: The conjugate metric isd(x; y) = jjxy 1 jj = d(e; xy 1 ) = d(x 1 ; y 1 ): Let A = Auth(T ) denote the set of bounded autohomeomorphisms h from T to T (i.e.…”

“…In Section 2 below we give the CET, in what we call its conjuction form (the motivation being the need to handle bilateral shifts t z m ; t + z m ): In Section 3 we work in normed groups, as in [BOst12], extending the bitopological approach of [BOst-bit] to this more general setting. What motivates such a broader context is the re-interpretation of a sequence of self-homeomorphisms h n (t) uniformly converging to the identity as giving rise to null function sequences z n (t) := h n (t) t (converging in supremum norm to zero) which need not be constant as in the KBD Theorem.…”