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Let $N$ be a positive integer. We want to find a completely multiplicative functions f(n) with values $\pm 1$ for $n \le N$ such that the discrepancy $$D=\max_{n \le N} |\{\sum_{i=1}^nf(i)\}|$$ is as small as possible. This is Erdős Discrepancy problem for multiplicative functions.

Consider the following greedy algorithm:

After you assigned the values $f(2),f(3),\dots f(p_i)$ for the first $i$ primes assign the value $f(p_{i+1})$ so as to minimize the maximum discrepancy $|\{\sum_{i=1}^nf(i)\}|$ in every partial sum where unassigned entries of $f$ get the value zero.

Question: How does this greedy algorithm perform?

Experimental or heuristic answers as well as rigorous proofs are welcomed.

For more background and related questions see this post .

Variation

Consider the same greedy algorithm when you impose the condition that $f(m)=0$ unless $m$ is square free. (If $m$ is not square free $f$ is multiplicative and has values $\pm1$.)

Question: How does our greedy algorithm performs on the square-free version?

Namely, we would like to understand the behavior of the discrepancy of the function obtained by our greedy algorithm. While for EDP there are known examples with $\log N$ discrepancy, this is not known for the square-free version.

Update:

The very nice answer by rlo suggests that the greedy algorithm gives discrepancy close to $n^{1/3}$ or so, and rlo expect it also for the square free variation. Can an upper bound of $N^{1/2-\epsilon}$ be proved? What about a lower bound of $N^{\epsilon}$. Another interesting question is if you can improve the greedy algorithm to get lower discrepancy. Our greedy ignore 0's in intervals. A greedy algorithm that ignore intervals with 0's was considered in polymath5 and to the best of my memory achieve discrepancy $n^{1/2}$. Maybe a clever interpolation between these two variants will do a better job than both?

Further meditation and a new variant

It seems that in our greedy algorithm the decisions we make for small primes are fairly irrelevant. A way to check it:

Run the algorithm for N and test what is the discrepancy for an interval [1,T] where T is, say, $\sqrt N$. I would expect the answer to be roughly $\sqrt T$.

So now we can think about the following variation:

Let $a>1$ be a real number. We run the greedy algorithm above but our decision for $f(p)$ is based only on intervals $[1,n]$ where $n \le p^a$. (Of course we consider only $n \le N$.

Questions: Can this variant lead to lower discrepancy?

What is the optimal value of $a$?

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Right sorry I meant completely multiplicative. –  Gil Kalai Aug 24 '12 at 13:20
2  
Are we requiring $f(n)$ to be $\pm 1$ only? –  rlo Aug 24 '12 at 15:03
    
yes. (We can also think of a variant with complex values of norm 1) –  Gil Kalai Aug 24 '12 at 15:58
    
It is still unclear. Suppose I start by assigning fK2), and propogate the result to all powers of 2. I get two infinite vectors of partial sums. How do I determine which vector I prefer so that I decide which choice I like? Do I consider all values, or just finitely many? Gerhard "Even Large Numbers Concern Me" Paseman, 2012.08.24 –  Gerhard Paseman Aug 24 '12 at 16:30
1  
@Gerhard Paseman: You fix some $N$. Then run the algo to get some function (up to $N$). Then compute its discrepancy. How will these discrepancies behave (as a function of $N$). –  quid Aug 24 '12 at 16:59

2 Answers 2

up vote 11 down vote accepted

Update 2: Original answer below. I've put together graphs showing more than just champions, using every $N\leq 10^4$ and also every $N\equiv 0\pmod{100}$ up to $10^5$. This is for the original version, not the variant, but I'd expect that to be essentially the same. I am starting to be somewhat skeptical of my $1/3$ estimate. I'll collect some data further out, but it'll be less complete since it's more costly to collect.

Here's the graphs of $D(N)$ versus $N$. The added curves are $N^{1/3}$ and $\log N$. enter image description here enter image description here

Here's the graph of $\log D/\log N$ versus $N$. The horizontal line is at $1/3$. (Note the different scales.) enter image description here enter image description here

Original answer: I have some basic numerical observations. I hacked together some code in c++ to work on this, and would be happy to collect more focused data or to share the code. Also, whenever there was a choice of whether to assign the value $+1$ or $-1$ to $f(p)$, I chose $-1$ for consistency of output. This yields a well-defined function $f_N$ for each $N$. Let $D(N)$ denote the discrepancy of $f_N$ up to $N$.

(I've also looked at choosing $f(p)=+1$ if it's undetermined, and at $f(p)=\pm 1$ according to whether $p\equiv 1,3\pmod{4}$. In these cases, the data is essentially the same as below.)

  1. $D(N)$ is roughly increasing, but is not monotonic. Its champion values for $N\leq10000$ are, in the form $(N,D(N))$, $(1,1)$,$(10,2)$,$(24,3)$,$(70,5)$,$(91,6)$,$(391,7)$,$(553,8)$,$(668,9)$,$(961,10)$,$(1235,11)$,$(1265,13)$,$(2561,14)$,$(2604,17)$,$(6275,18)$,$(6276,19),\dots$. This growth is more than logarithmic, and is probably polynomial -- $\log D/\log N$ hovers pretty close to $1/3$ for each of these points, so that may be the answer for the $\Omega$ result.

  2. It appears that the functions $f_N$ may converge as $N\to\infty$, but I'm not sure of this and need more data. Let $l(N)$ denote the least prime $p$ for which the value of $f_N(p)$ is undetermined. Certainly $l(N)\geq 5$ once $N\geq 4$, and while there is a great deal of fluctuation, it appears that maybe $l(N)\geq 7$ once $N\geq40500$; I will be seeing if this is (numerically) true.

I hope to update this answer once I have more data.

Update: For the variation where we only look at the values of $f_N$ on squarefree integers, the behavior appears to be the same.

$(N,D(N),\log D/\log N)$:

  • $(1,1,\text{NaN})$
  • $(30,2,0.2037950471)$
  • $(42,3,0.2939297479)$
  • $(77,4,0.3191428313)$
  • $(190,5,0.3067334722)$
  • $(238,6,0.3274252273)$
  • $(319,8,0.3606890916)$
  • $(939,9,0.3210056698)$
  • $(1358,10,0.3191931033)$
  • $(1461,11,0.3290703914)$
  • $(2185,13,0.3335707591)$
  • $(2769,14,0.3329519195)$
  • $(3354,15,0.3335896252)$
  • $(3689,17,0.3449622741)$
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After when $f_N(p)$ the function is periodic. Thus $f_N(p)$ does not depend on $N$ for $N>2\cdot3\cdots p$, and so $f(N)$ converges. It may be hard to compute the limit thought. –  Omer Sep 23 '12 at 17:30
    
The links seem to have gone away. –  Gerry Myerson Mar 7 at 5:19
1  
Fixed! Sorry about that. Apparently all the files I'd hosted at Emory before moving have gone away. –  rlo Mar 7 at 17:08

Here is a plot of rlo's data (as listed in his/her item (1)):
          Plot D(N)


And here is the $\log D(N) / \log N$ ratio he mentioned in the comment (where the constant red line is $\frac{1}{3}$):
          Plot LogD/LogN

share|improve this answer
1  
His. And the ratio of $\log D/\log N$ I alluded to, not in graph form: 0.3010299957, 0.345687124, 0.3788254007, 0.3972102138, 0.3260186782, 0.3302216042, 0.3378116896, 0.3352640758, 0.3368385727, 0.3590944053, 0.336264762, 0.3602395359, 0.3305424391, 0.3367194215. Close enough to 1/3 that I suspect something is happening. –  rlo Aug 25 '12 at 0:30

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