Incomplete Additive Character Sums and Applications Arne Winterhof Institut f¨ ur Diskrete Mathematik, ¨ Osterreichische Akademie der Wissenschaften, Sonnenfelsgasse 19, A–1010 Wien, Austria, arne.winterhof@oeaw.ac.at Dedicated to Klaus Burde on the occasion of his 65th birthday Abstract. We prove some bounds on incomplete additive character sums of poly- nomials over finite fields. We also apply results on incomplete additive character sums to get a distribution property of irreducible polynomials, we estimate the maximum running digital sums of some dc-constrained codes, and we describe a variant of Waring’s problem in finite fields. Keywords. Finite fields, exponential sums, incomplete character sums, dis- tribution of irreducible polynomials, constrained codes, Waring’s problem in finite fields 1 Introduction In the present article we consider incomplete additive character sums of poly- nomials ∑ x∈S ψ(g(x)), where S is a subset of the finite field F q , ψ the additive canonical character of F q , and g a polynomial over F q . In Section 2 we deduce a lower bound for arbitrary subsets S, that is an existence theorem, in Section 3 we prove upper bounds for some special S, and in Section 4 we describe some applications of results on incomplete ad- ditive character sums. In particular, we present a distribution property of irreducible polynomials, we investigate some dc-constrained codes, and we discuss a variant of Waring’s problem for finite fields. 2 A Lower Bound Theorem 1. Let n ≥ 2 and (n,q − 1) = 1. Then for any subset S ⊂ F q there exist b ∈ F ∗ q and a monic irreducible polynomial g ∈ F q [x] of degree n such that       x∈S ψ(bg(x))      ≥  |S|  |S|− 1 (q − 1) 2 +1  .