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We introduce the notion of set of uniqueness for $A(\\bar{D}^I)$ (respectively for $A(\\bar{B}_n)$) for compact subsets $K$ of $T^I$ (respectively of $\\partial \\bar{B}_n$) where $T=\\partial D$ is the unit circle. Our main result is that if $K$ has positive measure then $K$ is a set of uniqueness. The converse does not hold. 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Makridis, V. Nestoridis","submitted_at":"2013-04-19T19:28:05Z","abstract_excerpt":"We investigate the sets of uniform limits $A(\\bar{B}_n)$, $A(\\bar{D}^I)$ of polynomials on the closed unit ball $\\bar{B}_n$ of $\\mathbb{C}^n$ and on the cartesian product $\\bar{D}^I$ where $I$ is an arbitrary set and $\\bar{D}$ is the closed unit disc in $\\mathbb{C}$. We introduce the notion of set of uniqueness for $A(\\bar{D}^I)$ (respectively for $A(\\bar{B}_n)$) for compact subsets $K$ of $T^I$ (respectively of $\\partial \\bar{B}_n$) where $T=\\partial D$ is the unit circle. Our main result is that if $K$ has positive measure then $K$ is a set of uniqueness. The converse does not hold. 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