In a world where misinformation, voluntary or accidental, reigns supreme; in a world where lies become truth if they are broadcast for long enough; in a world where we have unlimited access to superintelligent machines, but we prefer to remain ignorant; in this world we are unfortunately living today, that is, the approach taken by scientists to accumulate knowledge - peer review - is something we should hold dear and preserve with care. And yet...
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- Last updated:
- April 03, 2025 02:25 AM
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, defined as the largest quantity such that it is possible to factorize 
into 
factors 
, each of which is at least 
, because on the one hand we can factor 
into nine factors, each of which is 
or higher.
; informally, this means we can split 
and transferring some powers of 
from the later part of the sequence to the earlier part to rebalance the terms somewhat. More precisely, if one removes one power of two from the even numbers between 
and 
to 
powers of two that one can then distribute amongst the numbers up to 
to bring them all up to at least 
in size. A more complicated procedure involving transferring both powers of 
then gives the improvement 
. At this point, however, things got more complicated, and the following conjectures were made by Guy and Selfridge:
for all 
? 
for all 
? (At 
, this conjecture barely fails: 
.) 
for all 
?
is the explicit constant 
was previously conjectured 
singled out by Guy and Selfridge appears to be quite a suitable test case: the constructions I tried fell just a little short of the conjectured threshold of 
, but it seems barely within reach that a sufficiently efficient rearrangement of factors can work here.
between 
and 
divide it twice), and any factor 
that contains such a factor therefore has to be significantly larger than the benchmark value of 
) is what leads to the precise constant 
and then tried to “improve” this factorization by moving around some of the prime factors. For the lower bound in 
is the target lower bound (so, slightly smaller than 
is a moderately sized natural number parameter (we will take 
, although there is significant flexibility here). If we denote the right-hand side here by 
, then 
such factors, in fact). By some careful applications of the prime number theorem, one can tweak some of the large primes appearing in 
by the following procedure:
by 
. Thus 
. 
more 
powers of 
to each 
for a small 
(in most cases one can make 
, and often one also has 
). If we then replace 
for each 
of the 
powers of 
powers remaining at my disposal. (I don’t claim that this is the most efficient matching, in terms of powers of two required, but it sufficed.) 
, and then distributing the remaining 
powers of two arbitrarily, this obtains a factorization of 
in this manner. Perhaps some readers here who are adept with computers can come up with a more efficient construction to get closer to this bound? If one can find a way to reach this bound, most likely it can be adapted to then resolve conjectures (ii) and (iii) above after some additional numerical effort.
, which is also known as Archimedes’ constant: the 
really called the angle of declination?
, one can approximately write the relationship 
between the mountain height 
, the Earth radius 
, and the dip angle 
. The key point here is the inverse quadratic dependence on 
radians, leading to an estimate of 
and 
. If one takes the logarithm of the inverse square law in the video, and performs the normalizations used by astronomers to define magnitude, one arrives at the standard relation 
between absolute and apparent magnitude.
, known as the 
of the flatness of the universe which, for reasons still not fully understood (but likely caused by some sort of inflation mechanism), happens to be extremely close to zero. As such, predictions for the size of the universe remain highly volatile at the current level of measurement accuracy.
: that is, the semidirect product of 
In fact these give all the automorphisms of the group 
is class-preserving if for each 
there exists some 
such that
for every 
we call 
an inner automorphism. But some groups have class-preserving automorphisms that are not inner! These are the class-preserving outer automorphisms.
Greg Egan looked for automorphisms of the McGee group that are of the form
acting as automorphisms of an abelian group 
Then we can define the 
to be the group of 
-cocycles modulo 
here. A 1-cocycle is none other than a function 
obeying
You can check that every 1-coboundary is a 1-cocycle. 
is the group of 1-cocycles modulo 1-coboundaries.
and for any 
and suppose 
is a 1-cocycle
will have outer automorphisms if 
then 
for which
took its group of affine transformations 
and saw this group must have outer automorphisms if
and 
Furthermore, we found a class-preserving outer automorphism of 
in general?
?
for some increasing sequence 
of natural numbers. Of course, since most real numbers are irrational, one expects such series to “generically” be irrational, and we make this intuition precise (in both a probabilistic sense and a Baire category sense) in our paper. However, it is often difficult to establish the irrationality of any specific series. For example, it is already a non-trivial 
is irrational, while the irrationality of 
(equivalent to 
(
for any 
, then the Ahmes series is necessarily irrational, basically because the fractional parts of 
can be arbitrarily small positive quantities, which is inconsistent with 
to obtain a rational Ahmes series of growth rate 
for any fixed 
.
, for instance if 
for a fixed 
, then one can find a comparable sequence 
for which 
is rational. This partially addresses 
had this property, and whether any sequence of exponential or slower growth (but with 
convergent) had this property. Unfortunately we barely miss a full solution of both parts of the problem, since the condition 
with 
such that 
, although the second part (which asks about 
) is slightly out of reach of our methods. Indeed, we show that the exponential growth hypothesis is best possible in the sense a random sequence 
.
is rational for every rational 
to avoid division by zero)! This answers (in the negative) a question of Stolarsky 
as an Ahmes series 
(with the 
. More generally, to represent a tuple of numbers 
indexed by some set 
of numbers simultaneously as 
with 
, this is the same as asking for the infinite sumset 
be a Banach space, let 
be sets with each 
contained in the ball of radius 
around the origin for some 
with 
convergent, so that the infinite sumset 
is well-defined. Suppose that one has some convergent series 
in 
converging in norm to zero, such that 
. Then the infinite sumset 
. 
“at the scale 
“.
. Our task is to express 
as a series 
with 
. From 
and 
. Iterating this, we may find 
such that 
. Sending 
, we obtain 
are dense enough that the condition 
which will become dense enough to obtain usable inclusions of the form 
reaches the dimension of the ambient space, basically thanks to the inverse function theorem (and the non-vanishing curvatures of the curve in question). For the Stolarsky problem, which is an infinite-dimensional problem, it turns out that one can modify this approach by letting 
.
and 
—addressing directly the extent of topological protection of the states generated by non-Abelian anyons. Behavior clearly conforming to the trends highlighted above could certify the device as a topological quantum memory. (Personally, I most anxiously await this milestone.)
that contains a unit line segment in every direction, must have Minkowski and Hausdorff dimension equal to three. (There is also a stronger “maximal function” version of this conjecture that remains open at present, although the methods of this paper will give some non-trivial bounds on this maximal function.) It is common to discretize this conjecture in terms of small scale 
. Roughly speaking, the conjecture then asserts that if one has a family 
of 
tubes of cardinality 
, and pointing in a 
-separated set of directions, then the union 
of these tubes should have volume 
. Here we shall be a little vague as to what 
means here, but roughly one should think of this as “up to factors of the form 
for any 
“; in particular this notation can absorb any logarithmic losses that might arise for instance from a dyadic pigeonholing argument. For technical reasons (including the need to invoke the aforementioned dyadic pigeonholing), one actually works with slightly smaller sets 
, where 
is a “shading” of the tubes in 
of 
, that one would like to make as large as possible. For instance, just from considering a single tube in this collection, one can easily establish 
. By just using the fact that two lines in 
-based argument of Córdoba, one can obtain 
(and this type of argument also resolves the Kakeya conjecture in two dimensions). In 1995, building on earlier work by Bourgain, 
using what is now known as the “Wolff hairbrush argument”, based on considering the size of a “hairbrush” – the union of all the tubes that pass through a single tube (the hairbrush “stem”) in the collection.
. The proof is lengthy (127 pages!), and relies crucially on 
represent the assertion that 
for all 
is some small positive quantity depending on 
in a non-trivial fashion, with the hope that this non-trivial argument can somehow be perturbed or optimized to get the crucial improvement 
. The standard strategy for doing this, since the work of Bourgain and then Wolff in the 1990s (with precursors in older work of Córdoba), is to perform some sort of “induction on scales”. Here is the basic idea. Let us call the 
for some intermediate scale 
; it is not terribly important for this sketch precisely what intermediate value is chosen here, but one could for instance set 
if desired. Because of the 
thin tubes in a given fat tube, and so we need at least 
fat tubes to cover the 
of 
fat tubes 
, with each fat tube containing about 
of the thin tubes. Let us also assume that the fat tubes 
-separated in direction, which is an assumption which is highly consistent with the other assumptions made here.
, this morally tells us that the typical multiplicity 
of the fat tubes is 
; a typical point in 
should belong to about 
fat tubes.
to turn it into a 
tube, this would inflate the thin tubes to be rescaled tubes of dimensions 
, which would now be 
-separated in direction. This rescaling does not affect the multiplicity of the tubes. Applying 
of the rescaled tubes, and hence the thin tubes inside 
.
of the full collection 
tubes overall. This heuristically gives 
, which then recovers 
is something resembling what we might call a “Kakeya configuration” at scale 
thin tubes of dimension 
for some 
coming from 
is “non-sticky” (whatever that means).
fat tubes for some multiplicity 
(e.g., 
for some small constant 
), with each fat tube containing only 
thin tubes. Now we have an unfortunate imbalance: the fat tubes form a “super-Kakeya configuration”, with too many tubes at the coarse scale 
. The hypothesis 
of the form 
. Suppose for sake of argument that the set 
and some 
, and thus has volume 
by the hypothesis 
, this should imply that 
balls to cover it. Applying 
cubes. But it has been observed in several previous works (starting 
disjoint prisms (or “grains”) in various orientations for some intermediate scales 
. The original “graininess” argument of Nets, Izabella and myself required a stickiness hypothesis which we are explicitly not assuming (and also an “x-ray estimate”, though Wang and Zahl were able to find a suitable substitute for this), so is not directly available for this argument; however, there is an alternate approach to graininess 
. The grains associated to a single fat tube will be essentially disjoint; but there can be overlap between grains from different fat tubes.
of the grains are not specified in advance; the argument of Guth will show that 
is significantly larger than 
with 
much larger than 
; and for fixed 
with 
much larger than 
(i.e., 
), so that the Kakeya set 
is the multiplicity, not of the fat tubes 
. The point here is that by the graininess hypotheses, each 
. Inserting these bounds into 
which leads to the desired gain 
, with the grains in each prism 
. In this case, it turns out that the 
of an individual electron should solve the time-independent Schrödinger equation 
is the wave function of the electron at this energy level, and 
is the conduction band energy. The behavior of hole wavefunctions follows a similar equation, governed by the valence band energy 
instead of 
that comes from electrostatic effects from the electrons and holes, and more specifically by solving the Poisson equation
is the dielectric constant of the semiconductor, 
are the carrier densities of electrons and holes respectively, 
, 
are further densities of ionized acceptor and donor atoms, and 
are physical constants. This equation looks somewhat complicated, but is mostly determined by the carrier densities 
associated to the eigenfunctions 
is given by the formula
plays an important role in being able to solve the Schrödinger equation. (The Poisson equation, being elliptic, is relatively easy to solve to high accuracy by standard methods, such as finite element methods.) If the potential 
at various energy levels 
, defined to be the solution to the equation
an effective potential for the Schrödinger equation (and 
also being the coefficients of an effective geometry for the equation). In practice, when 
tends to be behave like a somewhat “smoothed out” or “homogenized” version of 
. This is accurate at very high energies 
for this density of states that is significantly more accurate in practice in these regimes, with a mathematical justification (up to multiplicative constants) of this accuracy obtained in 
