Exploring the sum-product conjecture

If we take products instead, how many distinct products are there?Proven resultsErdős and Szemerédi proved that there are constants c and ε > 0 such thatmax{|A+A|, |A*A|} ≥ c|A|1+εIn other words, either A+A or A*A is substantially bigger than A.

 Erdős and Szemerédi only proved that some positive ε exists, but they suspected ε could be chosen close to 1, i.

e.

that either |A+A| or |A*A| is bounded below by a fixed multiple of |A|² or nearly so.

 George Shakan later showed that one can take ε to be any value less than1/3 + 5/5277 = 0.

3342899…but the conjecture remains that the upper limit on ε is 1.

Python codeThe following Python code will let you explore the sum-product conjecture empirically.

It randomly selects sets of size N from the non-negative integers less than R, then computes the sum and product sets using set comprehensions.

from numpy.

random import choice def trial(R, N): # R = integer range, N = sample size x = choice(R, N, replace=False) s = {a+b for a in x for b in x} p = {a*b for a in x for b in x} return (len(s), len(p) When I first tried this code I thought it had a bug.

I called trial 10 times and got the same values for |A+A| and |A*A| every time.

That was because I chose R large relative to N.

In that case, it is likely that every sum and every product will be unique, aside from the redundancy from commutativity.

That is, if R >> N, it is likely that |A+A| and |A*A| will both equal N(N+1)/2.

Things get more interesting when N is closer to R.

Probability vs combinatoricsThe Erdős-Szemerédi problem is a problem in combinatorics, looking for deterministic lower bounds.

But it seems natural to consider a probabilistic extension.

Instead of asking about lower bounds on |A+A| and |A*A| you could ask for the distribution on |A+A| and |A*A| when the sets A are drawn from some probability distribution.

If the set A is drawn from a continuous distribution, then |A+A| and |A*A| both equal N(N+1)/2 with probability 1.

Only careful choices, ones that would happen randomly with probability zero, could prevent the sums and products from being unique, modulo commutativity, as in the case R >> N above.

If the set A is an arithmetic sequence then |A+A| is small and |A*A| is large, and the opposite holds if A is a geometric sequence.

So it might be interesting to look at the correlation of |A+A| and |A*A| when A comes from a discrete distribution, such as choosing N integers uniformly from [1, R] when N/R is not too small.

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