On the length of arithmetic progressions in linear combinations of S-units

Abstract. In this paper we consider representations of algebraic integers of a number field as linear combinations of units with coefficients coming from a fixed small set, and as sums of elements having small norms in absolute value. These theorems can be viewed as results concerning a generalization of the so-called unit sum number problem, as well. Beside these, extending previous related results we give an upper bound for the length of arithmetic progressions of t-term sums of algebraic integers having small norms in absolute value.

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“…The next result is Theorem 1.1 of Hajdu and Luca in [8]. For the first (non-explicit) result of this type see also [7].…”

Section: Proofs Of the Other Theoremsmentioning

confidence: 88%

Representing algebraic integers as linear combinations of units

2014

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“…is the constant appearing in the conclusion of Matveev's theorem (10) when Λ involves r = + 2 rational numbers (r = + 2 and D = 1). Now we show that…”

Section: T)mentioning

confidence: 99%

“…For related problems and results see e.g. the papers [1,2,5,10,11,12,18] and the references therein.…”

Section: Introductionmentioning

confidence: 99%