Gaps in sumsets of s pseudo s-th powers
Annales de l'Institut Fourier, Volume 67 (2017) no. 4, p. 1725-1738

We study the length of the gaps between consecutive members in the sumset sA when A is a pseudo s-th power sequence, with s≥2. We show that, almost surely, limsup(b n+1 -b n )/logb n =s s s!/Γ s (1/s), where b n are the elements of sA.

On étudie la taille des différences entre les termes consécutifs de la suite sA où A est une suite de pseudo-puissances s-ièmes avec s≥2. On montre qu’on a presque sûrement limsup(b n+1 -b n )/logb n =s s s!/Γ s (1/s), où les b n sont les éléments de la suite sA.

Received : 2016-02-21
Revised : 2016-07-07
Accepted : 2016-09-15
Published online : 2017-09-26
DOI : https://doi.org/10.5802/aif.3120
Classification:  11B83
Keywords: Additive Number Theory, Pseudo s-th powers, Probabilistic method 
@article{AIF_2017__67_4_1725_0,
     author = {Cilleruelo, Javier and Deshouillers, Jean-Marc},
     title = {Gaps in sumsets of $s$ pseudo $s$-th powers},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {67},
     number = {4},
     year = {2017},
     pages = {1725-1738},
     doi = {10.5802/aif.3120},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2017__67_4_1725_0}
}

Cilleruelo, Javier; Deshouillers, Jean-Marc. Gaps in sumsets of $s$ pseudo $s$-th powers. Annales de l'Institut Fourier, Volume 67 (2017) no. 4, pp. 1725-1738. doi : 10.5802/aif.3120. http://www.numdam.org/item/AIF_2017__67_4_1725_0/

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