Abstract We prove that if $f(n)$ is a Steinhaus or Rademacher random multiplicative function, there almost surely exist arbitrarily large values of $x$ for which $|\sum _{n \leq x} f(n)| \geq \sqrt{x} (\log \log x)^{1/4+o(1)}$. This the first such bound grows faster than $\sqrt{x}$, answering question Halász and proving conjecture Erd?s. It plausible exponent $1/4$ sharp in this problem. The pr...

where A,B ⊂ B(H), and the distance between A⊗Mn and B ⊗Mn is measured in B(H) ⊗ Mn ∼= B(H). We also investigate the consequences of “complete closeness”, i.e. what can be said when dcb(A,B) is small? For example, if dcb(A,B) is small, then any projection p ∈ A ⊗ Mn can be suitably approximated by a projection q ∈ B ⊗ Mn, leading an isomorphism K0(A) → K0(B) which maps [p]0 to [q]0. This strateg...

Linear arithmetic extended with free predicate symbols is undecidable, in general. We show that the restriction of linear arithmetic inequations to simple bounds extended with the Bernays-Schönfinkel-Ramsey free first-order fragment is decidable and NEXPTIME-complete. The result is almost tight because the Bernays-Schönfinkel-Ramsey fragment is undecidable in combination with linear difference ...