Unique differences in symmetric subsets of F P

Abstract

Let p be a prime and let A be a subset of F p with A = -A and |A \ {0}| ≤ 2log3(p). Then there is an element of F p which has a unique representation as a difference of two elements of A.

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References

  1. [1]

    J. Browkin, B. Divis and A. Schinzel: Addition of sequences in general fields, Monatsh. Math. 82 (1976), 261–268.

    MathSciNet  Article  MATH  Google Scholar 

  2. [2]

    E. Croot and T. Schoen: On sumsets and spectral gaps, Acta Arith. 136 (2009), 47–55.

    MathSciNet  Article  MATH  Google Scholar 

  3. [3]

    J. W. S. Cassels: On a conjecture of R. M. Robinson about sums of roots of unity, J. Reine Angew. Math. 238 (1969), 112–131.

    MathSciNet  MATH  Google Scholar 

  4. [4]

    K. Ireland and M. Rosen: A Classical Introduction to Modern Number Theory, Graduate Texts in Mathematics 84, Springer 1990.

    Google Scholar 

  5. [5]

    K. H. Leung and B. Schmidt: Unique Sums and Differences in Finite Abelian Groups, Submitted 1990.

    Google Scholar 

  6. [6]

    V. F. Lev: The rectifiability threshold in abelian groups, Combinatorica 28 (2008), 491–497.

    MathSciNet  Article  MATH  Google Scholar 

  7. [7]

    J. H. Loxton: On two problems of E. M. Robinson about sums of roots of unity, Acta Arith. 26 (1974), 159–174.

    MathSciNet  MATH  Google Scholar 

  8. [8]

    Z. Nedev: An algorithm for finding a nearly minimal balanced set in Fp, Math. Comp. 268 (2009), 2259–2267.

    Article  MATH  Google Scholar 

  9. [9]

    Z. Nedev: Lower bound for balanced sets, Theoret. Comput. Sci. 460 (2012), 89–93.

    MathSciNet  Article  MATH  Google Scholar 

  10. [10]

    Z. Nedev and A. Quas: Balanced sets and the vector game, Int. J. Number Theory 4 (2008), 339–347.

    MathSciNet  Article  MATH  Google Scholar 

  11. [11]

    C. Norman: Finitely Generated Abelian Groups and Similarity of Matrices over a Field, Springer 2012.

    Google Scholar 

  12. [12]

    B. Schmidt: Cyclotomic integers and finite geometry, J. Am. Math. Soc. 12 (1999), 929–952.

    MathSciNet  Article  MATH  Google Scholar 

  13. [13]

    E. G. Straus: Dierences of residues (mod p), J. Number Th. 8 (1976), 40–42.

    Article  MATH  Google Scholar 

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Correspondence to Bernhard Schmidt.

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Duc, T.D., Schmidt, B. Unique differences in symmetric subsets of F P . Combinatorica 37, 167–182 (2017). https://doi.org/10.1007/s00493-015-3282-9

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Mathematics Subject Classification (2000)

  • 11P70