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On the Erdős–Fuchs theorem

Volume 189 / 2019

Li-Xia Dai, Hao Pan Acta Arithmetica 189 (2019), 147-163 MSC: Primary 11P70; Secondary 11B13, 11B34. DOI: 10.4064/aa170724-11-7 Published online: 12 April 2019


We prove several extensions of the Erdős–Fuchs theorem. For example, for two subsets $A=\{a_1,a_2,\ldots\}$ and $B=\{b_1,b_2,\ldots\}$ of ${\mathbb N}$, if $$ a_i-b_i=o(a_i^{1/4}) $$ as $i\to \infty$, then $$ |\{(a,b): a\in A,\, b\in B,\, a+b\leq n\}|=cn+o(n^{1/4}) $$ is impossible for any constant $c \gt 0$.


  • Li-Xia DaiSchool of Mathematical Sciences
    Nanjing Normal University
    Nanjing 210046
    People’s Republic of China
  • Hao PanSchool of Applied Mathematics
    Nanjing University of Finance and Economics
    Nanjing 210046, People’s Republic of China

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