Erdös-Heilbronn problem

Let $ G $ be an Abelian group and let $ A \subset G $. For $ k \in \mathbf N $, let

$$ k \wedge A = \left \{ {\sum _ {x \in X } x } : {X \subset A \textrm{ and } \left | X \right | = k } \right \} . $$

(Here, $ | A | $ denotes the cardinality of a set $ A $.) Let $ p $ be a prime number and let $ A \subset \mathbf Z/p \mathbf Z $. It was conjectured by P. Erdös and H. Heilbronn that $ | {2 \wedge A } | \geq \min ( p,2 | A | - 3 ) $.

This conjecture, mentioned in [a5], was first proved in [a3], using linear algebra. As a consequence of the lower bound on the degree of the minimal polynomial of the Grasmann derivative, the following theorem is true [a3]: Let $ p $ be a prime number and let $ A \subset \mathbf Z/ {p \mathbf Z } $. Then

$$ \left | {k \wedge A } \right | \geq \min ( p,k ( \left | A \right | - k ) + 1 ) . $$

Applying this theorem with $ k = 2 $, one obtains the Erdös–Heilbronn conjecture mentioned above. A generalization of the theorem has been obtained in [a2]. Presently (1996), almost nothing is known for composite numbers. The following conjecture is proposed here: Let $ n $ be a composite number (cf. also Prime number) and let $ A \subset \mathbf Z/ {n \mathbf Z } $ be such that $ | A | \geq k - 1 + { {( n - 1 ) } / k } $. Then $ 0 \in j \wedge A $ for some $ 1 \leq j \leq k $.

For a prime number $ n $, the above conjecture is an easy consequence of the theorem above. Some applications to integer subset sums are contained in [a6]. Along the same lines, the conjecture has several implications. In particular, for $ k = 3 $ one finds: Let $ A \subset \{ 1 \dots n \} $ be such that $ | A | \geq 5 + { {( n - 1 ) } / 3 } $. Then there is a $ B \subset A $ such that $ 3 \leq | B | \leq 6 $ and $ \sum _ {x \in B } x = 2n $.

This was conjectured partially by Erdös and R. Graham [a5] and follows easily by applying the conjecture twice, after adding $ 0 $.

References

[a1]	N. Alon, "Subset sums" J. Number Th. , 27 (1987) pp. 196–205
[a2]	N. Alon, M.B. Nathanson, I. Z. Rusza, "The polynomial method and restricted sums of congruence classes" J. Number Th. 56, No. 2, 404-417 (1996) Zbl 0861.11006
[a3]	J.A. Dias da Silva, Y.O. Hamidoune, "Cyclic subspaces of Grassmann derivations" Bull. London Math. Soc. , 26 (1994) pp. 140–146
[a4]	P. Erdös, H. Heilbronn, "On the addition of residue classes mod $p$" Acta Arith. , 9 (1964) pp. 149–159
[a5]	P. Erdös, R. Graham, "Old and new problems and results in combinatorial number theory" L'Enseign. Math. (1980) pp. 1–128
[a6]	Y.O. Hamidoune, "The representation of some integers as a subset sum" Bull. London Math. Soc. , 26 (1994) pp. 557–563
[a7]	H.B. Mann, "Addition theorems" , R.E. Krieger (1976) (Edition: Second)