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      On a conjecture of Erd\H{o}s on additive basis of large orders : On a conjecture of Erd\H{o}s

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      research-article
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      AfricArXiv Preprints
      ScienceOpen
      circle of partition, axes, generalized circles of partition, generalized density
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            Abstract

            Using the methods of multivariate circles of partition, we prove that for any additive base $\mathbb{A}$ of order $h\geq 2$ the upper bound $$\# \left \{(x_1,x_2,\ldots,x_h)\in \mathbb{A}^h~|~\sum \limits_{i=1}^{h}x_i=k\right \}\ll_{h}\log k$$ holds for sufficiently large values of $k$ provided the counting function $$\# \left \{(x_1,x_2,\ldots,x_h)\in \mathbb{A}^h~|~\sum \limits_{i=1}^{h}x_i=k\right \}$$ is an increasing function for all $k$ sufficiently large.

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            Journal
            AfricArXiv Preprints
            ScienceOpen
            8 November 2022
            Affiliations
            [1 ] African Institute for Mathematical sciences
            Author notes
            Article
            10.14293/111.000/000050.v1
            e015ac65-16f1-4888-94be-c0a7d4234ce3

            This work has been published open access under Creative Commons Attribution License CC BY 4.0 , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Conditions, terms of use and publishing policy can be found at www.scienceopen.com .


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            Mathematics
            circle of partition,axes,generalized circles of partition,generalized density

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