### On the discrepancy of coloring finite sets.

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Various techniques are presented for constructing $\Lambda $ (p) sets which are not $\Lambda (p+\u03f5)$ for all $\u03f5\>0$. The main result is that there is a $\Lambda $ (4) set in the dual of any compact abelian group which is not $\Lambda (4+\u03f5)$ for all $\u03f5\>0$. Along the way to proving this, new constructions are given in dual groups in which constructions were already known of $\Lambda $ (p) not $\Lambda (p+\u03f5)$ sets, for certain values of $p$. The main new constructions in specific dual groups are: – there is a $\Lambda $ (2k) set which is not $\Lambda (2k+\u03f5)$ in $\mathbf{Z}\left(2\right)\oplus \mathbf{Z}\left(2\right)\oplus \cdots $ for all $2\le k$, $k\in \mathbf{N}$ and $\u03f5\>0$, and in...

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