19/19 You can find out more about this problem of Erdös (and many more other ones) by visiting the following page: erdosproblems.com/143.

18/19 Lastly, it should be noted that Erdös also asked if we can solve |a-nb|<ε under the weaker assumption that the sum of 1/(a*log a) over a in A diverges. Our method falls short of proving this. There is a key step that doesn't seem to adapt to this more general situation.

17/19 There were some other important complications when using the 2nd moment method: the events we were initially using had important positive correlations. We had to modify them using an old idea of Erdös from his work on primitive sets of integers.

16/19 There is a caveat though: our proof of IDAP uses ideas from a variant of the proof of DSC. This variant was developed recently by Hauke-Vazquez-Walker and it breaks one of the processes of the GCD method in two steps that allows for better control of some quantities.

15/19 The method of GCD graphs we developed to solve DSC turns out to be the key to locate the necessary "structure" in A, so that the "structure vs randomness" argument goes through.

14/19 This is very similar to the type of question we encountered with James Maynard when studying the Duffin-Schaeffer conjecture (DSC) in Diophantine approximation!

13/19 ...if a=m/n and b=m'/n', then the product GCD(m,m')*GCD(n,n') is "large" (in some appropriate sense). Could this then mean that there are many elements of A both of whose numerator and denominator have some large fixed divisor (which could account for large pairwise GCDs)?

12/19 Assume now that A is a given set of rational numbers that has positive upper log density, and yet there are no solutions to the inequality |a-nb|<ε. Using the 2nd moment method, we show that the only way this could happen is if there are many pairs (a,b) in AxA such that...

11/19 In the other extreme, we think of A as being "random" if all ratios a/b are irrational or perhaps rational numbers of appropriately large height.

10/19 We think of A as being "structured" if it is a set of integers, or a small perturbation of such a set. For example, A could consist of small chunks of the form {γm/n: m in [x_1,x_2]}, where γ is fixed real number and x_1,x_2 are two parameters.

9/19 In my work with Lamzouri and Lichtman, we prove the integer dilation approximation problem in full generality. Our proof uses a type of "structure vs randomness" strategy.

8/19 Other than the two above results, there were no results in any sort of "intermediate" case. For instance, here is a case that was open: what if A consists of rational numbers a=m/n whose numerators and denominators are rather large compared to the size of ρ (e.g. m>a^100)?

7/19 On the other hand, Haight worked in 1988 on a case of the problem that is, in a sense, completely orthogonal to the integer case: he proved that if for all distinct a,b in A the ratio a/b is an irrational number, then the integer dilation approximation problem is true for A.

6/19 But why are primitive sets not a counterexample to the IDAP? Because we know from work of Erdös and Behrend in the 1930s that primitive sets have zero logarithmic density! Hence, the integer dilation approximation problem follows in the case when A is a set of integers.

5/19 Indeed, we say that a set of integers A is primitive if for all distinct elements a and b of A, we have that b is NOT a multiple of a. In particular, for such a set, there are no solutions to the inequality |a-nb|<1 with a,b distinct elements of A and with n an integer.

4/19 We call this problem the integer dilation approximation problem (IDAP). When A is a set of integers and ε<1, the inequality |a-nb|<ε implies that a=nb, and thus b is a multiple of a. This is what connects the problem to primitive sets of integers.

3/19 Erdos thought that this would be possible as long as A has positive upper logarithmic density (this means that the sum of 1/a over all elements a of A lying in [1,x] is occasionally at least as big as c*log x, with c>0 a constant and with x tending to infinity).

2/19 Let A be a discrete set of positive real numbers and let ε>0. How large does A have to be in order for us to be able to guarantee we can find solutions to the inequality |a-nb|<ε with distinct a and b in A and with n an integer?