Today we talk about the paper, Proof of Tomaszewski’s Conjecture on Randomly Signed Sums, by Nathan Keller and Ohad Klein.
Consider a unit vector That is . Consider all () signed sums where each is either 1 or -1.
Theorem (Keller and Klein (2020) asked by Boguslav Tomaszewski (1986)): For at least signed sums
Another way to state the theorem is that the probability of a signed sum to be in the interval [-1, 1] is at least 1/2.
To see that this is best possible consider the case that and let be non zero. For the sum in question to exceed one we need both summands to have the same sign which happens half of the times. There is another example of importance, the vector . Here 3/8 of the absolute values of signed sums (6 out of 16) are below 1 (in fact, equal to zero), 1/2 equal to 1 and 1/8 exceed 1. Holzman and Kleitman proved in 1992 that the fraction of absolute values of signed sums below 1 is always at least 3/8.
Congratulations to Nathan and Ohad. I will say a little more about the problem below but before that, a few more things.
A few more things
Luca Trevisan posted on his blog In Theory a post “Silver linings” about two cheerful pieces of news. The first one is “Karlin, Klein, and Oveis Gharan have just posted a paper in which, at long last, they improve over the 1.5 approximation ratio for metric TSP which was achieved, in 1974, by Christofides.”
The second one is about breaking the logarithmic barrier for Roth’s theorem that we wrote about here. This was also discussed by Bill Gasarch on Computational Complexity. In the comment section of my post there is an interesting discussion regarding timetable for future achievements and how surprising they would be.
The third is about Ron Graham, a friend and a mathematical giant who passed away a few days ago. Here is a moving post by Fan Chung, a web page for Ron set by Fan, and a blog post by Dick and Ken on GLL.
The fourth is that there is a nice collection of open problems on Boolean functions that is cited in the paper of Nathan and Ohad: Y. Filmus, H. Hatami, S. Heilman, E. Mossel, R. O’Donnell, S. Sachdeva, A. Wan, and K. Wimmer, Real analysis in computer science: A collection of open problems.
The fifth is that both our (HUJI) combinatorics seminar and basic notions seminar are running and are recorded. Here are the links. (Hmm, the links are not yet available, I will update.)
Back to the result of Keller and Klein
Daniel Kleitman and Ron Holzman
A quick orientation
If the s are all the same, or small, or random, then to compute the probability that the weighted sum is between -1 and 1, we can use some Gaussian approximation and then we will find ourselves in a clash of constants that goes our way. The probability will be close to a constant well above 1/2. So what we need to understand is the case where some s are large.
Early papers on the problem
The problem first appeared in the American Math Monthly. Richard Guy collected several problems and challenged the readers Any Answers Anent These Analytical Enigmas? (I don’t know what the fate of the other questions is.) Holzman and Kleitman proved in 1992 that the fraction of absolute values of signed sums below 1 is always at least 3/8, and this is tight. For many years, 3/8 was the record for the original problem, until the 2017 paper by Ravi Boppana and Ron Holzman: Tomaszewski’s problem on randomly signed sums: Breaking the 3/8 barrier, where a lower bound of 0.406, was proved. The current record 0f 0.46 was proved in the paper Improved Bound for Tomaszewski’s Problem by Vojtěch Dvořák, Peter van Hintum, and Marius Tiba. The new definite result by Nathan and Ohad used some ideas of these early papers.
What is the crux of matters
Let me quote what the authors kindly wrote me:
“The crux of the matter is how to deal with the case of very large coefficients (). We gave a short semi-inductive argument covering this case (this is Section 5 of the paper). The argument is only semi-inductive, as it requires the full assertion of Tomaszewski for any n'<n, and gives only the case () for . But this means that if we can handle all other cases by other methods then we will be done.
The semi-inductive argument takes only 3 pages. Handling the other cases takes 72 more pages and requires several new tools, but is closer to things that were done in previous works. (Actually, after we found the 3-page argument, we were quite sure we will be able to finalize the proof; this indeed happened, but took a year).”
Most of the paper deals with the case of small coefficients. This requires several ideas and new tools.
Rademacher sums: Improved Berry-Esseen and local tail inequalities
If all coefficients are “sufficiently small”, then we can
approximate X by a Gaussian and the inequality should follow. However, using the standard Berry-Esseen bound, this holds only if all coefficients are less than 0.16.
Nathan and Ohad showed that for Rademacher sums, namely random variables of the form , as discussed in the conjecture, a stronger Berry-Esseen
bound can be obtained, and this bound shows immediately that Tomaszewski’s assertion holds whenever all coefficients are less than 0.31. The stronger bound stems
from a method of Prawitz, presented in the 1972 paper. H. Prawitz, Limits for a distribution, if the characteristic function is given in a finite domain, which appeared in the Scandinavian Actuarial journal.
The second tool is local tail inequalities for Rademacher sums, of the form where a,b,c,d satisfy certain conditions. Inequalities of this kind were obtained before by Devroye and Lugosi in the 2008 paper: Local tail bounds for functions of independent random variables.
These local tail inequalities already have some other applications, e.g., to analysis of Boolean functions. They were developed and applied in an earlier paper paper of Keller and Klein: Biased halfspaces, noise sensitivity, and relative Chernoff inequalities. Let me mention my related MO question A variance tail description for continuous probability distributions.
A couple more ingredients
A stopping time argument. Variants of Tomaszewski’s problem appeared in various fields. The problem was stated independently in a 2002 paper by Ben-Tal, Nemirovski, Roos, Robust solutions of uncertain quadratic and conic-quadratic problems. A stopping time argument introduced there (for proving a lower bound of 1/3) played a crucial role in subsequent works and the critical semi-inductive argument by Nathan and Ohad.
Refinements of the famous Chebyshev’s inequality. (Did you know Chebyshev’s full name? Ans: Pafnuty Lvovich Chebyshev.)
Questions and connections that come to mind
Q1: What can be said about families of signs that can serve as those signs for which for some vector .
Q2: What can be said about the complex version or even more generally about high dimensions?
Q3: Are there any relations to Littlewood-Offord type problems?
Q4: Is there any relation to the Komlos Conjecture?
Is there a simpler proof?
We can ask about simpler or just different proofs for almost every result we discuss here. But here the statement is so simple…