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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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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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