But I think that raises an interesting question. If we choose a large set of small segments of HAPs, then must the annihilator of that set consist of very structured functions (e.g. functions that are periodic with period for some small and zero at all multiples of )? Since this is a linear algebra question, it might be more approachable that EDP itself, but it also seems close enough that an answer to it could be helpful. Even if that is not the case, I like the question.

A few remarks:

1) Actually we do not have to determine in advance: we only need to assume we can partition the HAP to intervals of length proportional to k (say between k/100 and 100k) so that the equations will be satisfied. (We have a considerable freedom to chose the equations that will still to the discrepency being likely small.)

2) The linear algebra consideration can go both ways. If you can show using linear algebra that you cannot have the zeros of the sums on all HAP being too “close by” (say uniformly below some constant) this will come close of proving EDP.

(And as a meta comment: trying other techniques like linear algebra methods or polynomial methods or probabilistic methods seem quite reasonable for EDP. unlike DHJ i dont think we have a propri much knowledge which tricks will be relevant.)

3) Strangely we need to understand how such conditions on HAP with large periods depend on conditions for HAP with small periods; Both in terms of “statistical dependence” (my heuristic comment) and in terms of “linear algebra dependence” (your comment).

4) Also the problem of trying to get by stochastic methods low discrepency sequence has a feeling like random walks with various stochastic tools to make the walks frequently return to zero. (Simultaniously for all HAP). It seems we discussed such “biased” random walks before but I dont remember when.

5) Anyway these heuristics taking also with the linear algebra concerns still suggest that discrepency sqrt log n might be possible.

My guess is that this problem would be NP-complete in general, so the best one could hope for is a sufficient condition for such a function to exist. The motivation for the question is that one way of restricting the growth along a HAP is to insist, fairly frequently, that the sum is zero. This gives us a collection as above, and the question is how “dense” we can make it.

A small remark is that if we take the linear relaxation (that is, we allow to take arbitrary real values) then we always have the trivial solution of setting to be identically zero, and we have non-trivial solutions only if the characteristic functions of the sets are linearly independent. If we decide that the sum should be zero at every th number along each HAP, then the HAP with common difference gives rise to around linear conditions, so the total number of conditions we need is , which (for smallish ) is around . This suggests that there is something natural about the bound. (Of course, it is possible that there will be dependencies amongst the various conditions, but at first glance they look fairly independent to me.)

If my heuristic idea is roughly ok (and it is quite possible I made a serious mistake there), then I would expect that log n discrepency sequences (perhaos even logn^{1/2+epsilon)) will be a rare but positive-probability events, (the number of such sequences will still be exponential), so I would not believe some subtle negative correlation will interfere (of course, proving this is a different matter).

There is some little hope that positive correlations will somehow allow constant-discrepency sequences.

This also suggests that finding in an arbitrary +-1 sequence an interval in an HOP with large discrepency using a pobabilistic argument is something worth trying. But the big obstacle here (as before) would be how to distinguish a +-1 sequence from a 0 +1 -1 sequence.

One thing I didn’t understand was your assertion that the correlations were positive. For example, it looks as though by far the easiest way to make the discrepancy small along all HAPs with odd common difference is to take a periodic sequence such as 1, -1, 1, -1, … or 1, 1, -1, -1, 1, 1, -1, -1, … which suggests a negative correlation with low discrepancy along HAPs with common difference a power of 2.

It could be that this heuristic argument breaks down if one looks only at logarithmic discrepancy, but I still don’t see any evidence for positive correlation. Indeed, the rough picture I had in mind was that there was negative correlation, in that in general it seems that periodicity tends to help a lot with some common differences and be a big problem for others. Or perhaps I mean something a bit more complicated, like positive pairwise correlation but with negative correlations coming in when one tries to ensure many events all at once. (If EDP is true then something like this seems to be the case, since we probably do have reasonable pairwise correlations but we know that we can’t get all the events to hold simultaneously. But again I’m talking about bounded discrepancy rather than logarithmic discrepancy here.)

First, a remark: It looks that the only definite result in the positive direction is about functions correlated with a character and I would be happy if somebody can explain informally, “why” this result is true, and does it “explain” the observed log-n discrepency.

So I thought about probabilistic heuristical ideas towards sequences with low discrepency.

How to construct prove using a random method that sequences with low discrepency exist? or even to estimate roughly the number of such sequences? Heuristically, if we want to keep the discrepency on a AP below K we need to impose conditions on intervals of length about or a little less. (If we assume that then we can be pretty sure that the partial sums are below or so.)

Altogether on all HAP we have nlogn/(K^2) conditions; and each one represents an event whose probability is roughly 1/K, so a rough estimate of the overall probability for a sequence to satisfy all these conditions is (1/K)^nlogn/K^2 which should be larger than 2^-n.

So if K is little more than sqrt log n it looks good.

There are now good news and bad news:

The bad news is that we do not have independence. There are methods to prove the existence of rare events even if there is some slight dependencies (e.g., Lovasz Local Lemma) but you still need much more independence.

The good news is that it seems that the correlation is positive. So if you already assume that sums of s is zero (or close to zero) for certain entervals in some HAP then it is more likely that such sums are zero when you take sums along intervals in other HAP.

But the good news are not that good in the sense that I see no (even heuristically) reason to believe that this positive correlation will replace the 1/K with something larger than 1/2 and this will not make a big difference. (But this deserves more thinking.)

What about general AP (just as a sanity test). Here, the n log n is replaced by n^2 so K indeed is expected to be larger. It would be interesting to compare the discrepency results for AP (Roth?) with what the probabilistic heuristic gives to get some intuition.

If such a heuristic can show that the probability for a sequence with low discrepency (say polylog n) is sufficiently large, I see no reason why such low discrepency will force some periodic behavior.

For a pseudointeger we look at the density of pseudointegers that are divisible by x. In one of the models where EDP fails this is 0 for all and in another this is 1 for all x. I have proved, that if it is 0 for one x, it is 0 for all . If it is 1 for a , it must be 1 for all x.

It turns out, that for a set X of “pseudointegers” as I defined it here (@Tim: sorry for stealing the word) you can find a (non necessarily injective, but non-trivial) function with and . It is defined such that if the density of integers divisible by x is , then . If furthermore the limit

exists and we know that the density must exists.

I think it is a good idea to think of pseudointegers as subsets of R, but as Tim noticed here it gives some difficulties (if there exists n such that ), so it might still make sense to use the more general definition.

What I have been suggesting is an intermediate result, in which we assume not just a lower bound on but also an upper bound on . How might that help?

Well, if we knew that the only eigenvectors with very small eigenvalues were very trigonometric in shape, then we might be getting somewhere, since then the only way for to be large without correlating with a trigonometric function would be for to have a reasonable projection on to the space generated by eigenvectors with larger eigenvalues.

It’s quite plausible that low-eigenvalue eigenvectors are trigonometric in nature, since periodic examples seem to give us a rich supply of low-discrepancy sequences (and hence sequences that are mapped to something small by our matrix). So this might be something that’s not too hard to investigate.

Why isn’t this just restating the problem? I think it’s because I’d hope that we could find such a big supply of functions that are mapped to small things that we would “run out of the small part” of the spectral decomposition. It might not be necessary to find the eigenvectors. Sorry to be a bit vague here. I think the next thing to do is see what the matrix does to trigonometric functions.

Regarding the experimental evidence on prime squares vs thrice primes: isn’t this what we see happening based on the approximations here and here? For example, if is a thrice prime and if is a prime squared.

Admittedly, these approximations are (i) upper bounds and (ii) upper bounds on the behaviour of the first iterate of , rather than the eigenvector/fixed-point, but it seems an encouraging sign.

Incidentally, I think it’s also possible to provide lower bounds for this first iterate. For example if a prime, then , which looks like if is large. I haven’t yet done this in the cases or , but will try to do this at some stage soon unless you think this is a dead end?

Finally, another experimental observation: it also looks as though the primes cubed are worse than being twice a prime squared.

That is, we don’t care for partial sums, but only for HAPs that goes all the way to n.

Given a set of nodes :

1. One labelled set of edges connects to for all i from 1 to n-1.

2. For any edge from to , i is less than j.

3. Given a set consisting of all labelled edges of a given label, any traversal starting from the root node of the set can traverse the entire set of edges.

4. No set of edges is a subset of another set (this prevents having arbitrary start points).

Given these conditions, the minimal set of nodes where the discrepancy must be greater than 1 is 4.

Proof: Consider every possible set of edges using 3 nodes.

set A: 1->2->3
set B: 1->3

Then , , has a discrepancy of 1.

However, consider these sets with 4 nodes:

set A: 1->2->3->4
set B: 1->3
set C: 1->4

Then no combinations of +1 and -1 (set A constrains the possibilities to + – + -, – + + -, + – - +, and – + – +) avoids having a discrepancy of 2 in some set of edges.

The starting point is a wish to prove EDP by means of a suitable averaging argument. That is, instead of proving that there exists a HAP such that , one attempts to prove that the sum is large on average.

There is a great deal of flexibility in how one interprets this, since we can apply a monotonic function to the sum before taking the average, and we can also introduce weights. To be more explicit, suppose that is some strictly increasing function. Then if and only if . Therefore, if is a set of HAPs, and to each we assign a weight in such a way that , then a sufficient condition for there to exist a HAP with is that .

The most obvious choice of function is , since this is gives us a positive-definite quadratic form in and potentially makes available to us the tools of linear algebra.

We can think of this quadratic form as follows. Let be the linear map from to that’s defined by , and let be the diagonal map from to itself that has matrix . Then the quadratic form takes the column vector to . It ought to be useful if we could diagonalize this quadratic form, which is equivalent to finding functions such that . The most obvious way to try to do that is to take the symmetric matrix and find an orthonormal basis of eigenvectors. But it occurs to me while writing this that there might be other useful ways of diagonalizing the quadratic form. Or maybe one would be happy to settle for more than functions , as long as they were somehow “nice”. It is of course always open to us to choose the weights to try to make them nice.

Define an infinite set of nodes such that given a node there is a directed edge labelled from to for all .

Give each node a value 1 or -1. Consider a traversal starting at a node moving along edges with the same label a finite number of times; then the discrepancy of a given traversal is the absolute value of the sum of all nodes visited. Asking if it is possible for the discrepancy to be bounded is equivalent to our problem.

What’s interesting is the graph doesn’t have to follow the number system. Have a set of nodes (finite or infinite) such that there are labelled directed edges, but allow the edges to be arbitrary. Then there are interesting questions like; given a discrepancy d, what is a minimal graph (in terms of nodes and/or edges) where the discrepancy is greater than d? What conditions cause the discrepancy to “break”? My hope is we can isolate certain graph theoretic properties which cause the logic to break and translate those into number theoretic properties we can search for (perhaps very rare ones, but ones we can prove exist nonetheless).

A few experimental questions that interest me are as follows.

(i) What does the restriction of the graph of to numbers of the form look like?

(ii) More generally, if you pick a large prime and look at the ratios for small values of , then what does that look like as a function of ?

(iii) Does that function have more or less the same shape as itself?

(iv) What does look like at integers of the form with and primes of roughly the same size?

I’m still puzzled about why squares of primes should behave as they do. It seems that being a square of a prime is worse than being three times a prime, and much worse than being a for primes and of comparable size.

One thing I’m about to look at, without much hope of a miracle: is the eigenvector a multiplicative (as opposed to completely multiplicative) function?

There are some visible patterns: for example, the clearly defined upper stripe is the primes, and the most obvious stripe below is the twice-primes.