Geodesic
Sets in $\mathbb{Z}_m$ whose difference sets avoid squares
Sbornik. Mathematics, Tome 209 (2018) no. 11, pp. 1603-1610

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We prove that the bound $|A|\leq m^{1/2}(3n)^{1.5n}$ holds for all square-free $m\in\mathbb{N}$ and any set $A\subset\mathbb{Z}_m$ such that $A-A$ contains no nonzero squares, where $n$ denotes the number of odd prime divisors of $m$. Bibliography: 9 titles.
Keywords: This research was funded by a grant of the Russian Science Foundation (project no. 14-11-00702). 
@article{SM_2018_209_11_a2,
     author = {M. R. Gabdullin},
     title = {Sets in $\mathbb{Z}_m$ whose difference sets avoid squares},
     journal = {Sbornik. Mathematics},
     pages = {1603--1610},
     publisher = {mathdoc},
     volume = {209},
     number = {11},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2018_209_11_a2/}
}
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M. R. Gabdullin. Sets in $\mathbb{Z}_m$ whose difference sets avoid squares. Sbornik. Mathematics, Tome 209 (2018) no. 11, pp. 1603-1610. http://geodesic.mathdoc.fr/item/SM_2018_209_11_a2/