Geodesic
Romanoff additive theorem's proof and its analogues
Čebyševskij sbornik, Tome 17 (2016) no. 4, pp. 51-56.

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In paper we describe the way N. P. Romanoff proved his additive theorem and sufficient conditions to obtain its analogues for sets with similar distribution and arithmetic. Also the example of set with similar distribution but with different arithmetic is given. We prove that the Romanoff theorem's analogue for this set is incorrect. Bibliography: 9 titles.
Keywords: Romanoff theorem, sumset, exponential sums. 
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A. N. Vassilyev. Romanoff additive theorem's proof and its analogues. Čebyševskij sbornik, Tome 17 (2016) no. 4, pp. 51-56. http://geodesic.mathdoc.fr/item/CHEB_2016_17_4_a3/

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[2] Brun V., “‘Le crible d’Eratosthene et le theoreme de Goldbach”, C. R. Acad. Sci. Paris, 168 (1919), 544–546 | Zbl

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[7] Ballot C., Luca F., “On the sumset of the primes and a linear recurrence”, Acta Arithmetica, 161 (2013), 33–46 | DOI | MR | Zbl

[8] Pomerance C., “Divisors of the Middle Binomial Coefficient”, American Mathematical Monthly, 122 (2015), 636–644 | DOI | MR | Zbl

[9] Kummer E., “Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen”, Journal für die reine und angewandte Mathematik, 44 (1852), 93–146 | DOI | MR | Zbl