In orthodox first-order logic, variables and expressions are only allowed to take one value at a time; a variable 
However, the ability to allow expressions to become only partially specified is undeniably convenient, and also rather intuitive. A classic example here is that of the quadratic formula:
Strictly speaking, the expression 
or
or
in order to strictly adhere to this grammar. But none of these three reformulations are as compact or as conceptually clear as the original one. In a similar spirit, a mathematical English sentence such as
which are used once and then discarded.
Another example of partially specified notation is the innocuous 
Below the fold I’ll try to assign a formal meaning to partially specified expressions such as (1), for instance allowing one to condense (2), (3), (4) to just
or the little-o notation 
. We will explain how to do this at the end of this post.
, the 
, and the 
) in short intervals ![{(X,X+H]}](https://s0.wp.com/latex.php?latex=%7B%28X%2CX%2BH%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
, where 
is large and 
lies in the range 
for a fixed constant 
(that one would like to be as small as possible). If we let 
denote one of the functions 
, then there is extensive literature on the estimation of short sums 
, where 
. For applications in the additive combinatorics of such functions 
is a polynomial of some fixed degree, or more generally 
is a nilmanifold of fixed degree and dimension (and with some control on structure constants), 
is a polynomial map, and 
is a Lipschitz function (with some bound on the Lipschitz constant). Indeed, thanks to the inverse theorem for the Gowers uniformity norm, such correlations let one control the Gowers uniformity norm of 
would be approximated in terms of a main term that vanished (or is negligible) if 
from each of the arithmetic functions 
variable to be restricted further to a subprogression of 
, as per the Möbius pseudorandomness conjecture. (One could choose a more sophisticated approximant in the presence of a Siegel zero, as I did with Joni in 
, where 
and 
. 
for some explicit polynomials 
, chosen so that 
and 
over the trivial bound for any 
; for 
over the trivial bound for some 
. In the case of the Möbius function 
somewhat at the cost of only obtaining “weakly logarithmic savings” of shape 
for some small 
.
, one can obtain strongly logarithmic savings on 
, and power savings for 
. 
, one can obtain weakly logarithmic savings for 
. 
, one can obtain power savings for 
. 
, one can obtain power savings for 
.
was established 
was established previously by ![{[1,X]}](https://s0.wp.com/latex.php?latex=%7B%5B1%2CX%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
. 
into more tractable sums of the following types:
sums, which are basically of the form 
for some weights 
of controlled size and some cutoff 
that is not too large; 
sums, which are basically of the form 
for some weights 
of controlled size and some cutoffs 
that are not too close to 
or to 
sums, which are basically of the form 
for some weights 
depend on the choice of 
is basically of the form 
, and the “non-abelian” case in which 
is non-abelian and 
exhibits non-trivial oscillation in a central direction. In the abelian case we can adapt 
for some Dirichlet character 
and some frequency 
, but in this case one can use classical multiplicative methods to control the correlation. It turns out that the non-abelian case can be treated similarly. After applying Cauchy-Schwarz, one ends up analyzing the equidistribution of the four-variable polynomial sequence 
range in various dyadic intervals. Using the known multidimensional equidistribution theory of polynomial maps in nilmanifolds, one can eventually show in the non-abelian case that this sequence either has enough equidistribution to give cancellation, or else the nilsequence involved can be replaced with one from a lower dimensional nilmanifold, in which case one can apply an induction hypothesis.
into a number of arithmetic progressions, and then uses equidistribution theory to establish cancellation of sequences such as 
on the majority of these progressions. As it turns out, this strategy works well in the regime 
unless the nilsequence involved is “major arc”, but the latter case is treatable by existing methods as discussed previously; this is why the 
result can be as low as 
.
![{(x,x+H]}](https://s0.wp.com/latex.php?latex=%7B%28x%2Cx%2BH%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
with ![{[X,2X]}](https://s0.wp.com/latex.php?latex=%7B%5BX%2C2X%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
, in which we will be able to lower 
.
, but now we consider the more general situation of tiling a measure space 
shifted by a finite subset 
that acts in a measure-preserving (or at least quasi-measure-preserving) fashion on 
, 
, acting on 
, 
of 
, and refer to this as a measurable tiling of 
, we can create a measurable tiling with ![{A = [0,1/2]^2 \hbox{ mod } {\bf Z}^2}](https://s0.wp.com/latex.php?latex=%7BA+%3D+%5B0%2C1%2F2%5D%5E2+%5Chbox%7B+mod+%7D+%7B%5Cbf+Z%7D%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002)
and 
. Our main results are the following:
, whenever 
is an integer coprime to all primes up to the cardinality 
of 
of 
of the one-dimensional torus 
are rational, in the sense that 
. This answers a recent conjecture 
of higher-dimensional tori, the tiling need not be rational. However, we can show that we can “slide” the tiling to be rational by giving each translate 
of 
, and for every time 
still form a partition of 
modulo null sets, and at time 
the tiling becomes rational. In particular, if a set 
one can arrange matters so that all the velocities 
are parallel. If we furthermore assume that the tile 
form a 
-invariant subset of the torus.
of a finitely generated discrete group 
, with 
. (Nonabelian local tilings, for instance of the sphere by rotations, are of interest due to connections with the Banach-Tarski paradox; see the aforementioned 
implies 
or 
for real numbers 
– expressed through the medium of German-language rap:
What's new 
Recent Comments