On the Erdős–Fuchs theorem
Volume 189 / 2019
Acta Arithmetica 189 (2019), 147-163
MSC: Primary 11P70; Secondary 11B13, 11B34.
DOI: 10.4064/aa170724-11-7
Published online: 12 April 2019
Abstract
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 $c \gt 0$.
