Representing integers as sums or differences of general power products

Bertók, Csanád

doi:10.1007/s10474-013-0309-9

Cited by 3 publications

(1 citation statement)

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“…Questions such as: does every integer have a multi-base representation, or: what is the smallest number that cannot be represented in a certain system, are also of great interest; cf. [4,5,6,19]. The number of multi-base representations has also been analyzed; see [17,18].…”

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On the minimal Hamming weight of a multi-base representation

Krenn

Suppakitpaisarn

Wagner

2020

Journal of Number Theory

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Given a finite set of bases b1, b2, . . . , br (integers greater than 1), a multi-base representation of an integer n is a sum with summands db α 1 1 b α 2 2 · · · b αr r , where the αj are nonnegative integers and the digits d are taken from a fixed finite set. We consider multi-base representations with at least two bases that are multiplicatively independent. Our main result states that the order of magnitude of the minimal Hamming weight of an integer n, i.e., the minimal number of nonzero summands in a representation of n, is log n/(log log n). This is independent of the number of bases, the bases themselves, and the digit set.For the proof, the existing upper bound for prime bases is generalized to multiplicatively independent bases; for the required analysis of the natural greedy algorithm, an auxiliary result in Diophantine approximation is derived. The lower bound follows by a counting argument and alternatively by using communication complexity; thereby improving the existing bounds and closing the gap in the order of magnitude. This implies also that the greedy algorithm terminates after O(log n/ log log n) steps, and that this bound is sharp.

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On the minimal Hamming weight of a multi-base representation

Krenn

Suppakitpaisarn

Wagner

2020

Journal of Number Theory

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Limit Points of Nathanson’s Lambda Sequences

Singh

2021

Springer Proceedings in Mathematics &Amp; Statistics

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Unit Equations in Diophantine Number Theory

Evertse

Győry

2015

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Diophantine number theory is an active area that has seen tremendous growth over the past century, and in this theory unit equations play a central role. This comprehensive treatment is the first volume devoted to these equations. The authors gather together all the most important results and look at many different aspects, including effective results on unit equations over number fields, estimates on the number of solutions, analogues for function fields and effective results for unit equations over finitely generated domains. They also present a variety of applications. Introductory chapters provide the necessary background in algebraic number theory and function field theory, as well as an account of the required tools from Diophantine approximation and transcendence theory. This makes the book suitable for young researchers as well as experts who are looking for an up-to-date overview of the field.

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Representing integers as sums or differences of general power products

Cited by 3 publications

References 10 publications

On the minimal Hamming weight of a multi-base representation

On the minimal Hamming weight of a multi-base representation

Limit Points of Nathanson’s Lambda Sequences

Unit Equations in Diophantine Number Theory

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