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Abstract

We construct examples of Λ(4) sets E ⊂ ℤ. The construction uses certain families of thin intervals ${I_k} = ℒ ≈ E. The Λ(4) property for E is obtained from the stronger result that $||f||_{4} ≤ c||(∑|f_{I_k}|^2)^{1/2}_||_{4}$ where $f̂$ is supported on $(⋃ I_k) ⋂ ℤ$ and $f_{I_k}$ is defined by $f̂_{I_k} = χ_{I_k}f̂$. The proof of the latter involves a graph structure defined in terms of ℒ × ℒ (which is essentially E × E).