Geodesic
Difference Sets and Positive Exponential Sums. II: Cubic Residues in Cyclic Groups
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analytic and Combinatorial Number Theory, Tome 314 (2021), pp. 145-151

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By constructing suitable nonnegative exponential sums, we give upper bounds on the cardinality of any set $B_q$ in cyclic groups $\mathbb Z_q$ such that the difference set $B_q-B_q$ avoids cubic residues modulo $q$. 
@article{TM_2021_314_a7,
     author = {M\'at\'e Matolcsi and Imre Z. Ruzsa},
     title = {Difference {Sets} and {Positive} {Exponential} {Sums.} {II:} {Cubic} {Residues} in {Cyclic} {Groups}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {145--151},
     publisher = {mathdoc},
     volume = {314},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2021_314_a7/}
}
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Máté Matolcsi; Imre Z. Ruzsa. Difference Sets and Positive Exponential Sums. II: Cubic Residues in Cyclic Groups. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analytic and Combinatorial Number Theory, Tome 314 (2021), pp. 145-151. http://geodesic.mathdoc.fr/item/TM_2021_314_a7/