A class of -sets
A continuous Helson surface in
For some time it has been known that there exist continuous Helson curves in . This result, which is related to Lusin’s rearrangement problem, had been proved first by Kahane in 1968 with the aid of Baire category arguments. Later McGehee and Woodward extended this result, giving a concrete construction of a Helson -manifold in for . We present a construction of a Helson 2-manifold in . With modification, our method should even suffice to prove that there are Helson hypersurfaces in any .
A further note on a class of -sets
A generalization of a theorem of Erdős-Rényi to m-fold sums and differences
Let m ≥ 2 be a positive integer. Given a set E(ω) ⊆ ℕ we define to be the number of ways to represent N ∈ ℤ as a combination of sums and differences of m distinct elements of E(ω). In this paper, we prove the existence of a “thick” set E(ω) and a positive constant K such that for all N ∈ ℤ. This is a generalization of a known theorem by Erdős and Rényi. We also apply our results to harmonic analysis, where we prove the existence of certain thin sets.
A Helson set of uniqueness but not of synthesis
In [3] I showed that there are Helson sets on the circle 𝕋 which are not of synthesis, by constructing a Helson set which was not of uniqueness and so automatically not of synthesis. In [2] Kaufman gave a substantially simpler construction of such a set; his construction is now standard. It is natural to ask whether there exist Helson sets which are of uniqueness but not of synthesis; this has circulated as an open question. The answer is "yes" and was also given in [3, pp. 87-92] but seems to...
A new group algebra and lacunary sets in discrete noncommutative groups
A note on Helson sets
A Note on L-sets
Answering a question of Pisier, posed in [10], we construct an L-set which is not a finite union of translates of free sets.
Addendum to “On Hilbert sets and -spaces with no subspace isomorphic to ”
An application of Grothendieck's inequality to a problem in harmonic analysis
An extremal set of uniqueness?