Rainbow solutions of a linear equation with coefficients in Z/pZ

• Autores: Mario Huicochea
• Localización: Discrete Mathematics Days 2022 / Luis Felipe Tabera Alonso (ed. lit.), 2022, ISBN 978-84-19024-02-2, págs. 153-158
• Idioma: inglés
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• Resumen

  • Let p be a prime, n ∈ Z+ and w ∈ (0, 1). Given a colouring χ : Z/pZ → {1, 2, . . . , n}and a linear equationL : a1x1 + a2x2 + . . . + anxn = bwith a1, a2, . . . , an ∈ (Z/pZ)∗ and b ∈ Z/pZ fixed elements, we denote by R(χ,L) thefamily of vectors (b1, b2, . . . , bn) ∈ (Z/pZ)nsuch that a1b1 + a2b2 + . . . + anbn = b andχ−1(i)∩{b1, b2, . . . , bn} ̸= ∅ for each i ∈ {1, 2, . . . , n}. Here the main result is that there existsa constant c = c(w, n) > 0 with the following property: if χ is such that min1≤i≤n |χ−1(i)| ≥wp + 1 and if not all the coefficients a1, a2, . . . , an of L are equal, then|R(χ,L)| ≥ cpn−1.Moreover, this statement is sharp in different directions. A result about the solutions of Lin a grid is used in its proof and it is interesting in its own right.