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 Images of Additive Polynomials in $\FF_q ((t))$ Have the Optimal Approximation Property We show that the set of values of an additive polynomial in several variables with arguments in a formal Laurent series field over a finite field has the optimal approximation property: every element in the field has a (not necessarily unique) closest approximation in this set of values. The approximation is with respect to the canonical valuation on the field. This property is elementary in the language of valued rings. Categories:12J10, 12L12, 03C60