On the size of dissociated bases

1. On the size of dissociated bases Raphael YusterUniversity of HaifaJoint work with Vsevolod Lev University of Haifa

2. Dissociated bases • Recall, that subset sumsof a subset of an abelian group are group elements of the form: • where B • Note: there are at most 2||distinct subset sums. • Famous conjecture of Erdös(80 years ago, $500): • If all subset sums of an integer set  [1,n] are pairwise distinct, then || ≤ log n+O(1). • Similarly, one can investigate the largest possible size of subsets of other naturalsets in abelian groups, possessing the distinct subset sums property.

3. Example:What is the largest possible size of a set   {0,1}n n with all subset sums pairwise distinct? • Definition: a subset of an abelian group, all of whose subset sums are pairwise distinct, is called dissociated. • Dissociated sets are useful due to the fact that if is a maximal dissociated subset of a given set A, then every element of Ais representable as a linear combination of the elements of with the coefficients in {-1,0,1}. • Hence, maximal dissociated subsets of a given set can be considered as its ``linear bases” over the set {-1,0,1}. • This interpretation naturally makes one wonder whether, and to what extent, the size of a maximal dissociated subset of a given set is determined by this set?

4. Is it true that all maximal dissociated subsets of a given finite set in an abelian group are of about the same size? The following two theorems give a satisfactory answer:

5. Why is this a satisfactory answer: • Since the standard basis is a maximal dissociated subset of the set {0,1}n, comparing Theorems1 and 2 we conclude that: • Theorem 2is sharp in the sense that the logarithmic factors cannot be dropped or replaced with a slower growing function. • Theorem 1 is sharp in the sense that n log nis the true order of magnitude of the size of the largest dissociated subset of {0,1}n.

6. Outline of the proof of Theorem 1 • Recall: we want to prove that {0,1}npossesses a dissociated subset of size: • This is the same as showing that if n > (log 9+o(1))m/log mthen {0,1}npossesses an m-element dissociated subset. • The trick is to switch to the dual setting: • We prove that there exists a set D {0,1}mwith |D|=nsuch that for every non-zero vector s S:={-1,0,1}mthere is an element of D, not orthogonal to s. • Once this is done, we consider the nmmatrix whose rows are the elements of D; the columns of this matrix form an m-element dissociated subset of {0,1}n, as required.

7. Explanation: Suppose the sum of the red vectors is equal to the sum of the blue vectors. Then each row is orthogonal to the vector: • We construct Dby choosing uniformly at random, and independently of each other, n vectors from the set {0,1}m. • We will show that for every s S:={-1,0,1}m, the probability that all vectors from Dare orthogonal to sis very small. The rows are the elements of D  {0,1}m n m

8. We say that a vector from Sis of type (m+,m-) if it has m+ coordinates equal to +1, and m-coordinates equal to -1. • If sSis of type (m+,m-) then a vectord{0,1}mis orthogonal to sif and only if there exists j≥0 such that d has: - exactly jnon-zero coordinates in the (+1)-locations of s, - exactly jnon-zero coordinates in the (-1)-locations of s. • Example:s=(1,-1,0,1,1,-1) is of type (3,2) andd=(0,1 ,1,0,1,0 ) is orthogonal to s, here with j=1. • The probability for a randomly chosen d{0,1}mto be orthogonal to s is

9. It follows that the probability for alln elements of D to be simultaneously orthogonal to sis smaller than • Since the number of elements of a given type (m+,m-) isto conclude the proof it suffices to prove that • To this end we rewrite this sum asand split it into two parts, according to whether t<Tor t > T, where T := m/(log m)2. Denote the parts by ∑1 and ∑2 .

10. We prove that ∑1< ½and ∑2 < ½. • For ∑1 we have • As T := m/(log m)2 and n > (log 9+o(1))m/log m we haveand therefore ∑1< ½. • Proving that∑2< ½ is only slightly more involved.

11. Outline of the proof of Theorem 2 • Recall: we want to prove thatif  and M are maximal dissociated subsets of a subset A of an abelian group, then • Here we will only prove the lower bound: • The upper bound is only slightly more complicated. • By maximalityof , every element of A, and thereby every element M, is a linear combination of the elements of  with coefficients in {-1,0,1}. • Hence, every subset sum of Mis a linear combination of the elements of  with coefficients in {-|M|,-|M|+1,…,|M|}.

12. There are 2|M| subset sums of M, all distinct from each other. • There are (2|M|+1)||linear combinations of the elements of  with the coefficients in {-|M|,-|M|+1,…,|M|}. • we have: 2|M|·(2|M|+1)||and follows.

13. Open problem • For a positive integer n, let Lndenote the largest size of a dissociated subset of the set {0,1}n n. What are the limits • Notice, that by Theorems 1 and 2 we have

14. Thanks!