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When only the empty subsum is zero modulo \(p\). (Quand seule la sous-somme vide est nulle modulo \(p\).) (French) Zbl 1153.11007

A famous conjecture of Selfridge implies that any set \(S\) of residue classes modulo a prime \(p\) having no nonempty subset summing to zero must satisfy the condition \(|S|\leq\sqrt{2p}\). The author obtains an asymptotic solution for the last question using analytical tools.

MSC:

11B34 Representation functions
11P70 Inverse problems of additive number theory, including sumsets
11B50 Sequences (mod \(m\))
20D60 Arithmetic and combinatorial problems involving abstract finite groups

Keywords:

zero-sum subset
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References:

[1] Deshouillers J-M., A lower bound concerning subset sums which do not cover all the residues modulo \(p\). Hardy-Ramanujan J. 28 (2005), 30-34. · Zbl 1222.11037
[2] Deshouillers J-M., Freiman G. A., When subset-sums do not cover all the residues modulo \(p\). J. Number Theory 104 (2004), 255-262. · Zbl 1048.11077
[3] Erdős P., Heilbronn H., On the addition of the residue classes \(\text{mod}\, p\). Acta Arith. IX (1964), 149-159. · Zbl 0156.04801
[4] Olson J. E., An addition theorem modulo \(p\). J. Combin. Theory 5 (1968), 45-52. · Zbl 0174.05202
[5] Ould Hamidoune Y., Zémor G., On zero sum-free sets. Acta Arith. LXXVIII (1996), 143-152. · Zbl 0863.11016
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