Abstract
We construct a convex set A with cardinality 2n and with the property that an element of the difference set \(A-A\) can be represented in n different ways. We also show that this construction is optimal by proving that for any convex set A, the maximum possible number of representations an element of \(A-A\) can have is \(\lfloor |A|/2 \rfloor\).
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References
G. Elekes, M. Nathanson and I. Ruzsa, Convexity and sumsets, J. Number Theory, 83 (1999), 194-201.
M. Z. Garaev, On lower bounds for the \(L_1\)-norm of some exponential sums, Mat. Zametki, 68 (2000), 842-850 (in Russian); translation in Math. Notes, 68 (2000), 713-720.
S. V. Konyagin, An estimate for the \(L_1\)-norm of an exponential sums, in: Theory of Approximations of Functions and Operators, Abstracts of Papers of the International Conference, Dedicated to Stechkin's 80th Anniversary (Ekaterinbourg, 2000), pp. 88-89 (in Russian).
O. Raz, O. Roche-Newton and M. Sharir, Sets with few distinct distances do not have heavy lines, Discrete Math. 338 (2015), 1484-1492.
T. Schoen, On convolutions of convex sets and related problems, Canad. Math. Bull., 57 (2014), 877-883.
T. Schoen and I. Shkredov, On sumsets of convex sets, Combin. Probab. Comput., 20 (2011), 793-798.
I. Shkredov, `Some new results on higher energies', Trans. Moscow Math. Soc. 2013, 31-63.
Acknowledgements
We are grateful to Brandon Hanson, Misha Rudnev and Dmitrii Zhelezov for helpfully sharing their insights. We are particularly grateful to Ilya Shkredov for informing us about the reference [5].
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The authors were supported by the Austrian Science Fund FWF Project P 34180.
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Roche-Newton, O., Warren, A. A convex set with a rich difference. Acta Math. Hungar. 168, 587–592 (2022). https://doi.org/10.1007/s10474-022-01286-3
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DOI: https://doi.org/10.1007/s10474-022-01286-3