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When subset-sums do not cover all the residues modulo \(p\). (English) Zbl 1048.11077

If \({\mathcal A}\) is a subset of some group, \(S_{\mathcal A}\) denotes the sum of the elements of \({\mathcal A}\) and \({\mathcal A}^*= \{S_{\mathcal B},\,{\mathcal B}\subset{\mathcal A}\}\).
In [Acta Arith. 9, 149–159 (1964; Zbl 0156.04801)], P. Erdős and H. Heilbronn proved that every subset \({\mathcal A}\subset\mathbb{Z}/p\mathbb{Z}\setminus \{0\}\) such that \(|{\mathcal A}|\geq 3\sqrt{6p}\) satisfies \({\mathcal A}^*= \mathbb{Z}/p\mathbb{Z}\). They conjectured that the conclusion still holds if we replace the coefficient \(3\sqrt{6}\) by \(2\). Olson proved it and finally, Dias da Silva and ould Hamidoune proved a best possible result of this form.
The present paper looks for subsets \({\mathcal A}\subset \mathbb{Z}/p\mathbb{Z}\) such that \(|{\mathcal A}|\) is large and \({\mathcal A}^*\neq \mathbb{Z}/p\mathbb{Z}\).
Observe that as soon as \(\sum_{a\in{\mathcal A}}\| a/p\|< 1-1/p\) \((\| z\|= \min_{n\in\mathbb{Z}}| z-n|)\), \({\mathcal A}^*\neq \mathbb{Z}/p\mathbb{Z}\) must hold. More generally, if for some non-zero modulo \(p\) integer \(s\), \(\sum_{a\in{\mathcal A}}\| sa/p\|< 1-1/p\), then \({\mathcal A}^*\neq \mathbb{Z}/p\mathbb{Z}\) must hold.
The present paper looks for a kind of converse statement for this criterion. The following theorem is proved: Let \(c> \sqrt{2}\) and \(p\) be a prime. Let \({\mathcal A}\subset \mathbb{Z}/p\mathbb{Z}\), with \(|{\mathcal A}|\geq c\sqrt{p}\). If we assume \({\mathcal A}^*\neq \mathbb{Z}/p\mathbb{Z}\) then \(\sum_{a\in{\mathcal A}}\| as/p\|< 1+ O(p^{-1/4}\log p)\) for some integer \(s\) that is, non-zero modulo \(p\).

MSC:

11P70 Inverse problems of additive number theory, including sumsets
11B25 Arithmetic progressions

Citations:

Zbl 0156.04801
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Full Text: DOI

References:

[1] Chaimovich, M., New algorithm for dense subset sum problem, Astérisque, 258, 363-373 (1999) · Zbl 0987.90061
[2] Dias da Silva, J. A.; Hamidoune, Y. O., Cyclic spaces for Grassman derivatives and additive theory, Bull. London Math. Soc., 26, 140-146 (1994) · Zbl 0819.11007
[3] Erdős, P.; Heilbronn, H., On the addition of residue classes mod \(p\), Acta Arithmetica, 9, 149-159 (1964) · Zbl 0156.04801
[4] Freiman, G. A., New analytical results in subset sum problem, Discrete Math., 114, 205-217 (1993) · Zbl 0849.11015
[5] Galil, Z.; Margalit, O., An almost linear-time algorithm for the dense subset sum problem, SIAM J. Comput., 20, 6, 1157-1189 (1991) · Zbl 0736.68041
[6] Olson, J., An addition theorem modulo \(p\), J. Combin. Theory, 5, 45-52 (1968) · Zbl 0174.05202
[7] Sárközy, A., Finite addition theorems II, J. Number Theory, 48, 2, 197-218 (1994) · Zbl 0808.11011
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