A UNIFORM SET WITH FEWER THAN EXPECTED ARITHMETIC

PROGRESSIONS OF LENGTH 4.

W. T. GOWERS

Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK

Dedicated to Endre Szemerédi on the occasion of his 80th birthday.

Abstract. An example is constructed of a subset $A\subset{{Z}}_{N}$ of density
${1\over 2}+o(1)$ such that all the non-trivial Fourier coefficients of the
characteristic function of $A$ are very small, but if $x,d\in{{Z}}_{N}$ are
chosen uniformly at random, then the probability that $x,x+d,x+2d$ and $x+3d$
all belong to $A$ is at most ${1\over 16}-c$, where $c>0$ is an absolute
constant.

1\. Introduction.

The purpose of this short note is to disprove a conjecture made in [3], which
gave a proof of Szemerédi’s theorem [5] that generalized to longer
progressions Roth’s analytic approach [4] to the case of progressions of
length 3. Let us very briefly explain the motivation for the conjecture, and
the reason that it seemed plausible: more details can be found in [3].

The main guiding principle behind the proof of [3], and indeed all other known
proofs of Szemerédi’s theorem, is that a set that is sufficiently “random-
like” contains about as many arithmetic progressions of length $k$ as a
typical random set of the same density. However, it is not immediately obvious
what should count as a “random-like” set. If one is interested in progressions
of length 3, then a good definition turns out to come from Fourier analysis.
First, one associates with a subset $A\subset\\{1,2,\dots,N\\}$ its
characteristic function. Following [3], we shall write $A(x)$ rather than
$\chi_{A}(x)$. Next, one regards this function as a function defined on
${{Z}}_{N}\equiv{{Z}}/N{{Z}}$ so that there is a group structure appropriate
for discrete Fourier analysis. Having done so, one defines a discrete Fourier
transform as follows. If $f:{{Z}}_{N}\rightarrow{{C}}$ then

${\hat{f}}(r)=\mathop{{{E}}}_{x\in{{Z}}_{N}}f(x)\omega^{-rx}\ ,$

where $\omega=\mathop{\rm exp}(2\pi i/N)$, and $\mathop{{{E}}}$ is a standard
abbreviation for $N^{-1}\sum$.

Note that ${\hat{f}}(0)=\mathop{{{E}}}_{x}f(x)$, so if we return to our set
$A$, then ${\hat{A}}(0)=|A|/N$, or in other words the density of $A$. It turns
out that an appropriate definition of quasirandomness for progressions of
length three is that ${\hat{A}}(r)$ should be significantly smaller than this
for all non-zero $r$. To be precise, one can say that $A$ is a $c$-uniform set
if $|{\hat{A}}(r)|\mathchar 13366\relax c$ for every non-zero $r$.

A well-known and straightforward calculation shows that if $A$ has density
$\alpha$ and is $c$-uniform for $c$ significantly smaller than $\alpha^{2}$,
then

$\mathop{{{E}}}_{x+z=2y}A(x)A(y)A(z)\approx\alpha^{3}.$

That is, the “3-AP density” of $A$ is roughly $\alpha^{3}$, which is roughly
what it would be (with high probability) for a random set of density $\alpha$.
(Note that here we are counting mod-$N$ arithmetic progressions.)

What makes it hard to prove Szemerédi’s theorem analytically is that a set can
be $c$-uniform for a very small $c$, even one that tends to zero quite fast as
$N$ tends to infinity, but yet not contain roughly the expected number of
arithmetic progressions of length 4. An example was given in [3], which we
shall discuss in a moment. However, a defect of the set defined in that
example was that the number of progressions of length 4 was greater than one
would expect for a random set, so it did not demonstrate in a wholly
satisfactory way that $c$-uniformity could not be used. At one stage, I
thought it would be routine to convert the example into one for which the
number of progressions was smaller, but eventually Gil Kalai asked me how I
proposed to do it, and I realized that it was not routine after all. In fact,
all the obvious ideas seemed to fail, and the result was the conjecture in
[3], which says (in its mod-$N$ version) that if $A$ is a set of density
$\delta$ that is $c$-uniform, then the number of quadruples in $A^{4}$ of the
form $(x,x+d,x+2d,x+3d)$ is at least $(\delta^{4}-c^{\prime})N^{2}$, where
$c^{\prime}$ tends to zero as $c$ tends to zero. It is this conjecture that we
shall disprove here.

The example from [3] that yields too many progressions of length 4 is easy to
define. Fix a small constant $c$, such as $1/1000$, and let
$A\subset{{Z}}_{N}$ be the set of all $x\in{{Z}}_{N}$ such that the least
residue of $x^{2}$ mod $N$ lies in the interval $[-cN,cN]$. It is not too hard
to verify that $A$ is $N^{-1/2+\epsilon}$-uniform (which, since
$\sum_{r}|{\hat{A}}(r)|^{2}\mathchar 13366\relax 1$ is almost as uniform as a
set can possibly be) and has cardinality approximately $2cN$: proofs of these
facts will be sketched later in this paper. Therefore, by the argument above,
$A$ contains about $8c^{3}N^{2}$ triples $(x,x+d,x+2d)$. In fact, one can say
a little more: triples of the form $(x^{2},(x+d)^{2},(x+2d)^{2})$ are
“approximately uniformly distributed” in the sense that, given any three
reasonably long intervals $I$, $J$ and $K$, they contain about
$N^{-1}|I||J||K|$ of them. Now we use this uniform distribution property,
together with the identity

$(x+3d)^{2}=x^{2}-3(x+d)^{2}+3(x+2d)^{2}$

to conclude that, if $x$ and $d$ are chosen randomly, then the probability
that $(x+3d)^{2}$ belongs to $[-cN,cN]$ given that $x^{2}$, $(x+d)^{2}$ and
$(x+2d)^{2}$ all belong to $[-cN,cN]$ is significantly bigger than $2c$
(indeed, it will be approximately equal to an absolute constant that one could
work out after a tedious calculation), and therefore that the number of
quadruples $(x,x+d,x+2d,x+3d)$ in $A^{4}$ is significantly greater than
$16c^{4}N^{2}$.

A natural way to try to generalize this example is by replacing the interval
$[-cN,cN]$ with a more general set $B\subset{{Z}}_{N}$. For the above argument
to work, we would want $B$ to be a union of longish intervals, so that the
uniform distribution property was still valid. Then we could define $A$ to be
$\\{x\in{{Z}}_{N}:x^{2}\in B\\}$. If $B$ has size $\beta N$, then the uniform
distribution property implies that the number of triples in $A$ of the form
$(x,x+d,x+2d)$ is approximately $\beta^{3}N^{2}$, and that the corresponding
triples $(x^{2},(x+d)^{2},(x+2d)^{2})$ are “approximately uniformly
distributed” in $B^{3}$. We now want to estimate the conditional probability
that $(x+3d)^{2}\in B$, given that $x^{2}$, $(x+d)^{2}$ and $(x+2d)^{2}$ all
lie in $B$.

If the distribution were genuinely uniform, then, by the identity we used
earlier, this conditional probability would be $|B|^{-3}$ times the number of
quadruples $(x,y,z,w)\in B^{4}$ such that $w=x-3y+3z$. The notion of uniform
distribution turns out to be strong enough for this to be a good estimate even
in the situation we actually face, so let us think about how many such
quadruples there are, by expressing the number in terms of Fourier
coefficients. We obtain

$\eqalign{\mathop{{{E}}}_{x-3y+3z-w=0}B(x)B(y)B(z)B(w)&=\sum_{r}\sum_{x,y,z,w}B(x)B(y)B(z)B(w)\omega^{r(x-3y+3z-w)}\cr&=\sum_{r}|{\hat{B}}(r)|^{2}|{\hat{B}}(3r)|^{2}\cr&=\beta^{4}+\sum_{r\neq
0}|{\hat{B}}(r)|^{2}|{\hat{B}}(3r)|^{2}\cr}$

where $\beta=|B|/N$ is the density of $B$. Therefore, the conditional
probability in question is at least $\beta$, since the contribution from the
non-zero $r$ is positive. Therefore, the number of progressions of length 4 in
$A$ is bounded below (at least approximately) by $\beta^{4}N^{2}$.

It is not necessary to use Fourier analysis to obtain this conclusion: one can
also observe that the left-hand side is the square of the $L_{2}$ norm of the
convolution of $B$ with a dilate of $B$, and apply Cauchy-Schwarz. In any
case, the conclusion shows that no $A$ constructed in this way can give an
example of a set with too few progressions of length 4.

Note that the above argument depends on the positivity of the Fourier
expression, which in turn depends on the fact that the set $1,-3,3,-1$ can be
partitioned into pairs of the form $\\{a,-a\\}$. The non-Fourier argument also
relies on this fact, in a slightly different way. Thus, the fact that 4 is an
even number is highly relevant. Indeed, this obstruction does not exist for
odd $k$: it is possible to choose a union of intervals $B$ of density $\beta$
such that the number of quintuples $(v,w,x,y,z)\in B^{5}$ with
$v-4w+6x-4y+z=0$ is significantly less than $\beta^{5}N^{5}$. Then one can
define $A$ to be $\\{x\in{{Z}}_{N}:x^{3}\in B\\}$.

This set can be shown to have small Fourier coefficients. It turns out that
the uniform distribution property applies to quadruples of the form
$(x^{3},(x+d)^{3},(x+2d)^{3},(x+3d)^{3})$, from which it follows that $A$
contains fewer than $c\beta^{5}N^{2}$ quintuples of the form
$(x,x+d,x+2d,x+3d,x+4d)$, where $c<1$ is an absolute constant. The details of
this argument were worked out in a conversation with Ben Green, but here we
leave them as an exercise for the interested reader and return to progressions
of length 4.

We have now given the main motivation for the false conjecture in [3]: there
seems to be a clear difference between 4 and 5, arising from their differing
parities; for progressions of length 4, there is a certain positivity
phenomenon that stops examples of a certain kind from having too few
progressions of length 4. However, with a little more ingenuity one can after
all produce an example that works for progressions of length 4, as we shall
now show.

2\. Construction of a uniform set with few progressions of length 4.

###### Lemma 1

There is a function $f:{{Z}}\rightarrow\\{-1,0,1\\}$ such that $f(n)=0$ unless
$1\mathchar 13366\relax n\mathchar 13366\relax 300$ and such that

$\sum_{x,d\in{{Z}}}f(x)f(x+d)f(x+2d)f(x+3d)<0\ .$

Proof. First let us define a function $g:{{Z}}^{3}\rightarrow\\{-1,0,1\\}$
with the required property (where now $x$ and $d$ are elements of ${{Z}}^{3}$)
that is supported in the set $\\{1,2,3,4\\}^{3}$. Given
$(a,b,c)\in\\{1,2,3,4\\}^{3}$ we let $g(a,b,c)=1$ unless $(a,b,c)$ is one of
the following sixteen triples, in which case we set $g(a,b,c)=-1$:
113,121,132,144,212,224, 233,241,314,322,331,343,411,423,434 and 442\. These
triples are chosen to have the following property: every line that goes
through four points of the grid $\\{1,2,3,4\\}^{3}$ and is not one of the four
main diagonals contains precisely one of these points. (The search for such a
system of points was not very difficult, since in each horizontal plane one
had to take exactly one point in each row and column, and this had to be done
in a disjoint way for the four planes. This was already enough of a
restriction to make aesthetically driven trial and error a feasible method.)

Each pair $(x,d)$ that contributes to the sum above, with $g$ instead of $f$,
is either degenerate, in the sense that $d=0$, or it corresponds to a
geometrical line. Note also that each non-degenerate line is counted twice,
since the line for $x$ and $d$ is the same as the line for $x+3d$ and $-d$.
The contribution to the sum from each degenerate pairs is $1$, and there are
$4^{3}=64$ of them. The contribution from each line that is parallel to one of
the three coordinate axes is $-2$ and there are $3\times 16=48$ of these. The
contribution from each non-main diagonal is $-2$ and there are $3\times 8=24$
of those. Finally, each of the four main diagonals contributes $2$. Therefore,
the sum is $64-96-48+8=-72$.

Now we transfer this example to ${{Z}}$ by a suitable projection. Define a map
$\phi:\\{1,2,3,4\\}^{3}\rightarrow{{Z}}$ by $\phi(a,b,c)=a+8b+64c$. This map
is easily checked to be a Freiman homomorphism: that is,
$\phi(x)-\phi(y)=\phi(z)-\phi(w)$ if and only if $x-y=z-w$. Indeed, if

$a_{1}+8b_{1}+64c_{1}-a_{2}-8b_{2}-64c_{2}=a_{3}+8b_{3}+64c_{3}-a_{4}-8b_{4}-64c_{4}\
,$

then

$(a_{1}-a_{2}-a_{3}+a_{4})+8(b_{1}-b_{2}-b_{3}+b_{4})+64(c_{1}-c_{2}-c_{3}+c_{4})=0\
.$

But $a_{1}-a_{2}-a_{3}+a_{4}$, $b_{1}-b_{2}-b_{3}+b_{4}$ and
$c_{1}-c_{2}-c_{3}+c_{4}$ all lie between $-6$ and $6$, from which it follows
easily that they are all zero.

A quadruple of the form $(x,x+d,x+2d,x+3d)$ is the same as a quadruple
$(x,y,z,w)$ such that $y-x=z-y$ and $z-y=w-z$. Therefore, $(\phi x,\phi y,\phi
z,\phi w)$ is such a quadruple if and only if $(x,y,z,w)$ is. It follows that
if we define $f(\phi(a,b,c))$ to be $g(a,b,c)$ and $f(x)$ to be 0 if $x$ is
not in the image of $\phi$, then

$\sum_{x,d}f(x)f(x+d)f(x+2d)f(x+3d)=-72\ .$

This proves the lemma. $\mathchar 1027\relax$

Remark. A more efficient version of the above lemma was obtained by Wolf [6],
who found a function $f$ supported on $\\{1,2,\dots,18\\}$ that takes only
$\pm 1$ values in that range, for which the sum is -36.

###### Corollary 2

There exists an absolute constant $c>0$ such that for every sufficiently large
$N$ there is a function $F:{{Z}}_{N}\rightarrow\\{-1,0,1\\}$ with the
following properties. First, $F$ is a $\pm 1$-combination of characteristic
functions of disjoint intervals of size at least $N/1500$, and, secondly,

$\mathop{{{E}}}_{x,d\in{{Z}}_{N}}F(x)F(x+d)F(x+2d)F(x+3d)\mathchar
13366\relax-c\ .$

Proof. Let $t$ be a positive integer between $N/1500$ and $N/1200$, and for
$1\mathchar 13366\relax k\mathchar 13366\relax 300$ let $I_{k}$ be the
interval $\\{(2k-1)t+1,(2k-1)t+2,\dots,2kt\\}$. Then an argument similar to
the proof that the function $\phi$ in Lemma 1 was a Freiman homomorphism shows
that if $x$ and $d$ are elements of ${{Z}}_{N}$ and $x\in I_{k}$, $x+d\in
I_{l}$, $x+2d\in I_{m}$ and $x+3d\in I_{n}$, then $l-k=m-l=n-m$. In other
words, an arithmetic progression can lie in the union of the intervals $I_{k}$
only if the corresponding intervals themselves lie in an arithmetic
progression (which may be degenerate).

Suppose that $(k,l,m,n)$ is an arithmetic progression. Then the number of
arithmetic progressions $(x,y,z,w)\in I_{k}\times I_{l}\times I_{m}\times
I_{n}$ is the same as the number of such progressions in
$\\{1,2,\dots,t\\}^{4}$, since the map

$(x,y,z,w)\mapsto(x-(2k-1)t,y-(2l-1)t,z-(2m-1)t,w-(2n-1)t)$

is a bijection between the two sets of progressions in question. Notice also
that every mod-$N$ progression that lives in the union of the $I_{k}$ must in
fact correspond to a genuine arithmetic progression in the set
$\\{1,2,\dots,N\\}$, since the union does not contain any numbers between
$N/2$ and $N$.

Let $p$ be the number of progressions in $\\{1,2,\dots,t\\}^{4}$, where by
“progression” we mean “quadruple of the form $(x,x+d,x+2d,x+3d)$” and allow
$d$ to be any element of ${{Z}}_{N}$. Then $p\mathchar 13374\relax t^{2}/3$,
since if we choose two points at random in $\\{1,2,\dots,t\\}$ there is a
probability at least ${1\over 3}$ that they differ by a multiple of 3.

Now let $F$ be defined as follows. If $x\in I_{k}$ then $F(x)=f(k)$. For all
other $x$, $F(x)=0$. The remarks we have just made show that

$\mathop{{{E}}}_{x,d\in{{Z}}_{N}}F(x)F(x+d)F(x+2d)F(x+3d)=pN^{-2}\sum_{x,d\in{{Z}}}f(x)f(x+d)f(x+2d)f(x+3d)\
.$

But $p\mathchar 13374\relax t^{2}/3$, $t\mathchar 13374\relax N/1500$ and
$\sum_{x,d\in{{Z}}}f(x)f(x+d)f(x+2d)f(x+3d)=-72$. Therefore, the lemma is
proved, with $c=72/3(1500)^{2}$, which is greater than $1/100000$. $\mathchar
1027\relax$

Corollary 2 may seem rather pointless, since the function $F$ is composed of
characteristic functions of intervals and therefore has large Fourier
coefficients. However, the main observation that underlies the construction is
that such an example can be converted into a uniform set in a rather simple
way, while keeping its property of having a negative sum over progressions of
length 4.

The trick is to multiply $F$ pointwise by a small linear combination of
“quadratic phase functions,” that is, functions of the form $\omega^{q(x)}$
for a quadratic polynomial $q$. These have two properties that make them
useful. First, quadratic functions satisfy the identity

$q(x)-3q(x+d)+3q(x+2d)-q(x+3d)=0\ ,$

as we have already noted. Secondly, functions of the form $\omega^{q(x)}$ are
highly uniform – indeed, so uniform that they remain highly uniform even when
multiplied by characteristic functions of intervals. Let us prove the second
of these facts, which will allow us to show that the function we create is
uniform. The proof is standard, but is included for convenience. Once we have
established the second fact, we can show why the first fact helps us to obtain
a good estimate for sums of products over arithmetic progressions.

###### Lemma 3

Let $I$ be an interval of length $t$ in ${{Z}}_{N}$ and let $q$ be a quadratic
polynomial. Then no Fourier coefficient of the product $I(x)\omega^{q(x)}$ is
greater than $2N^{-1/2}\log N$.

Proof. First we show that all Fourier coefficients of the function
$Q(x)=\omega^{q(x)}$ have size $N^{-1/2}$. Indeed, let $q(x)=ax^{2}+bx+c$,
with $a\neq 0$. Then

$\eqalign{\Bigl{|}\mathop{{{E}}}_{x}Q(x)\omega^{-rx}\Bigr{|}^{2}&=\mathop{{{E}}}_{x,y}\omega^{-(q(x)-q(y)+r(x-y))}\cr&=\mathop{{{E}}}_{x,y}\omega^{-(ax+ay+b+r)(x-y)}\cr&=\mathop{{{E}}}_{z}\mathop{{{E}}}_{x}\omega^{-(2ax-
az+b+r)z}\ .\cr}$

For each fixed $z$, the average over $x$ is zero, except when $z=0$, in which
case it is $1$. Therefore, the average equals $N^{-1}$, which proves the
claim.

We now use the convolution identity in the less standard direction, which
tells us that the $r$th Fourier coefficient of the pointwise product $QI$ is
$\sum_{s+t=r}\hat{Q}(s)\hat{I}(r)$, which has modulus at most
$N^{-1/2}\sum_{r}|{\hat{I}}(r)|$.

By the formula for summing a geometric progression, we find that
$|\hat{I}(r)|$ is at most $2/N|1-\omega^{r}|$. One can also check that
$|1-\omega^{r}|\geq 4r/N$ when $-N/2\leq r\leq N/2$. It follows that
$\sum_{r}|{\hat{I}}(r)|$ is at most $1+\sum_{r\mathchar 13366\relax
N/2}r^{-1}$, which is at most $2\log N$. The result follows. $\mathchar
1027\relax$

Now let us define a function $G$ by

$G(x)=F(x)(\omega^{x^{2}}+\omega^{-x^{2}}+\omega^{3x^{2}}+\omega^{-3x^{2}})\
.$

Then, since $F$ is a $\pm 1$-combination of characteristic functions of $64$
intervals, Lemma 3 implies that no Fourier coefficient of $G$ exceeds $4\times
64\times 2N^{-1/2}\log N=512N^{-1/2}\log N$. Notice also that $G$ is real
valued and that its values all belong to the interval $[-4,4]$.

###### Lemma 4

Let $G$ be as just defined. Then

$\Bigl{|}\mathop{{{E}}}_{x,d}G(x)G(x+d)G(x+2d)G(x+3d)-2\mathop{{{E}}}_{x,d}F(x)F(x+d)F(x+2d)F(x+3d)\Bigr{|}$

is at most $2^{18}N^{-1/2}\log N$.

Proof. The product $G(x)G(x+d)G(x+2d)G(x+3d)$ splits into a sum of $256$ terms
of the form $\omega^{px^{2}+q(x+d)^{2}+r(x+2d)^{2}+s(x+3d)^{2}}$, where $p$,
$q$, $r$ and $s$ belong to the set $\\{-3,-1,1,3\\}$. Let us fix a choice of
$p$, $q$, $r$ and $s$, set
$\theta(x,d)=px^{2}+q(x+d)^{2}+r(x+2d)^{2}+s(x+3d)^{2}$, and estimate the
quantity

$\mathop{{{E}}}_{x,d}F(x)F(x+d)F(x+2d)F(x+3d)\omega^{\theta(x,d)}\ .$

We can write $\theta(x,d)=ux^{2}+vxd+wd^{2}$. Let us suppose that $u\neq 0$
and consider the average for a fixed value of $d$, writing $F_{d}(x)$ for
$F(x)F(x+d)F(x+2d)F(x+3d)$. The number of intervals $I_{k}$ on which $F$ is
non-zero is 64, so as $x$ increases, the number of times at least one of $x$,
$x+d$, $x+2d$ and $x+3d$ changes from not belonging to a certain $I_{k}$ to
belonging to it is at most $4\times 64=256$. It follows that $F_{d}$ is a $\pm
1$-sum of characteristic functions of at most $256$ disjoint intervals. Lemma
3 (in the case of the Fourier coefficient at zero) then implies that
$\mathop{{{E}}}_{x}F_{d}(x)\omega^{\theta(x,d)}$ has modulus at most
$512N^{-1/2}\log N$. If we now average over $d$ we find that

$\mathop{{{E}}}_{x,d}F(x)F(x+d)F(x+2d)F(x+3d)\omega^{\theta(x,d)}\mathchar
13366\relax 512N^{-1/2}\log N\ .$

Now let us consider what happens if $w\neq 0$. The argument is very similar.
This time we shall fix $x$ and define $F_{x}(d)$ to be
$F(x)F(x+d)F(x+2d)F(x+3d)$. (Note that we are using the same notation for a
different definition in this case.) As $d$ increases, the number of times
$x+rd$ can change from not belonging to $I_{k}$ to belonging to $I_{k}$ is at
most $r$. Therefore, $F_{x}$ is a $\pm 1$-sum of characteristic functions of
at most $(1+2+3)\times 64=384$ disjoint intervals. So in this case we obtain
the estimate

$\mathop{{{E}}}_{x,d}F(x)F(x+d)F(x+2d)F(x+3d)\omega^{\theta(x,d)}\mathchar
13366\relax 768N^{-1/2}\log N\ .$

If $u=0$ and $w=0$ then $p+q+r+s=0$ and $q+4r+9s=0$. Because $p$, $q$, $r$ and
$s$ all lie in the set $\\{-3,-1,1,3\\}$, it is not possible for $q+4r+9s$ to
equal 0 unless $s=\pm 1$. Since we can multiply any solution by $-1$ and
obtain another solution, let us suppose that $s=-1$. Then $q+4r=9$, which, it
is simple to check, can happen only if $q=-3$ and $r=3$. But then $p=1$. It
follows that the only two solutions are $(1,-3,3,-1)$ and $(-1,3,-3,1)$.

But in this case, $\theta(x,d)=0$, by the identity used earlier, so the
average becomes $\mathop{{{E}}}_{x,d}F(x)F(x+d)F(x+2d)F(x+3d)$. It follows
that the difference between the sum with $G$ and the sum with $F$ is at most
$256\times 1024N^{-1/2}\log N$, which is what the lemma states. $\mathchar
1027\relax$

We are almost done. $G$ is a function taking values in $[-4,4]$ with a
“negative number of progressions of length 4.” All that remains is to convert
it into a set. This we do by first setting $P(x)$ to be $(G(x)+4)/8$ and then
choosing a set $A$ by letting $x$ belong to $A$ with probability $P(x)$. Here
are the details.

###### Corollary 5

There exists an absolute constant $c>0$ such that for every sufficiently large
$N$ there is a function $P:{{Z}}_{N}\rightarrow[0,1]$ such that
$\Bigl{|}\mathop{{{E}}}_{x}P(x)-{1\over 2}\Bigr{|}\mathchar 13366\relax
64N^{-1/2}\log N$, such that $|{\hat{P}}(r)|\mathchar 13366\relax
64N^{-1/2}\log N$ for every $r\neq 0$ and such that

$\mathop{{{E}}}_{x,d}P(x)P(x+d)P(x+2d)P(x+3d)\mathchar 13366\relax{1\over
16}-c\ .$

Proof. As we have already said, we let $P(x)$ equal $(G(x)+4)/8$. Then
$|\mathop{{{E}}}_{x}P(x)-{1\over 2}|=|\mathop{{{E}}}_{x}G(x)/8|$, which is at
most $64N^{-1/2}\log N$, since it is ${\hat{G}}(0)/8$ and all Fourier
coefficients of $G$ are at most $512N^{-1/2}\log N$. Similarly, when $r\neq
0$, ${\hat{P}}(r)={\hat{G}}(r)/8$, since adding a constant to a function does
not alter its Fourier coefficients for $r\neq 0$.

To prove the last property, write the product $P(x)P(x+d)P(x+2d)P(x+3d)$ as

$2^{-12}(4+G(x))(4+G(x+d))(4+G(x+2d))(4+G(x+3d))\ .$

This product splits into 16 parts, each of which we shall average separately.

If we choose $4$ from every bracket, then we obtain $2^{8-12}={1\over 16}$ for
every $x$ and $d$. If we choose $G$ from every bracket then we are estimating

$2^{-12}\mathop{{{E}}}_{x,d}G(x)G(x+d)G(x+2d)G(x+3d)\ ,$

which, by Corollary 2 (with the quantitative estimate at the end) and Lemma 4
is at most $-2^{-12}.10^{-5}+2^{6}N^{-1/2}\log N$.

If we choose $G$ from precisely one bracket, then we obtain
$2^{-6}\mathop{{{E}}}_{x}G(x)$, after a suitable change of variables. We have
already remarked that $\left|\mathop{{{E}}}_{x}G(x)\right|\mathchar
13366\relax 512N^{-1/2}\log N$, so this is at most $8N^{-1/2}\log N$.

If we choose $G$ from precisely two brackets, then we obtain
$2^{-8}\left(\mathop{{{E}}}_{x}G(x)\right)^{2}$, again after a suitable change
of variables. This has modulus at most $1024N^{-1}(\log N)^{2}$.

If we choose $G$ from three brackets, then we must do four standard
calculations similar to the one that proves that a uniform set contains many
arithmetic progressions of length 3. We illustrate this by bounding the
quantity

$2^{-10}\mathop{{{E}}}_{x,d}G(x)G(x+d)G(x+3d)\ .$

First, a routine Fourier calculation shows that it is equal to

$\sum_{r}\hat{G}(2r)\hat{G}(-3r)\hat{G}(r)\ .$

But $|{\hat{G}}(r)|\mathchar 13366\relax 512N^{-1/2}\log N$ for every $r$.
Using this fact and the Cauchy-Schwarz inequality, the last expression can be
bounded above in modulus by

$512N^{-1/2}\log
N\Bigl{(}\sum_{r}|{\hat{G}}(2r)|^{2}\Bigr{)}^{1/2}\Bigl{(}\sum_{r}|{\hat{G}}(-3r)|^{2}\Bigr{)}^{1/2}\
.$

Since $N$ is prime, the product of the last two brackets is
$\|\hat{G}\|_{2}^{2}=\|G\|_{2}^{2}\leq 16$. Therefore, the whole sum is at
most $2^{13}N^{-1/2}\log N$. After multiplying by $2^{-10}$, as we need to, we
have shown that the contribution from this term is at most $8N^{-1/2}\log N$.
The estimates for the other terms with $G$ chosen three times are proved in a
very similar way.

There are sixteen terms, and the worst error term (when $N$ is sufficiently
large) has modulus at most $8N^{-1/2}\log N$. Therefore,
$\mathop{{{E}}}_{x,d}P(x)P(x+d)P(x+2d)P(x+3d)$ is at most
$(2^{-4}-2^{-12}.10^{-5})+(2^{6}+2^{7})N^{-1/2}\log N$. For sufficiently large
$N$, this is less than ${1\over 16}-2^{-29}$, which proves the corollary.
$\mathchar 1027\relax$

###### Theorem 6

There exist absolute constants $c>0$ and $C$ such that for all sufficiently
large $N$ there is a $CN^{-1/2}\log N$-uniform subset $A$ of ${{Z}}_{N}$ of
density ${1\over 2}+o(1)$, for which

$\mathop{{{E}}}_{x,d}A(x)A(x+d)A(x+2d)A(x+3d)\mathchar 13366\relax 2^{-4}-c\
.$

Proof. Choose $A$ randomly as follows. For every $x\in{{Z}}_{N}$, let $x$
belong to $A$ with probability $P(x)$, with all choices made independently.
Then $A(x)-P(x)$ is a random variable of mean zero that is equal to either
$1-P(x)$ or $-P(x)$. In particular, it always has modulus at most 1.

Now let us consider the difference between the Fourier coefficients of $A$ and
those of $P$. We have

${\hat{A}}(r)-{\hat{P}}(r)=\mathop{{{E}}}_{x}(A(x)-P(x))\omega^{rx}\ .$

This is an average of $N$ independent random variables, each of mean zero and
bounded above in modulus by 1. It follows immediately from Azuma’s inequality
that the probability that $|{\hat{A}}(r)-{\hat{P}}(r)|$ is greater than
$tN^{-1/2}$ is at most $2e^{-t^{2}/8}$. If we take $t$ to equal $\log N$, then
this probability is, for $N$ sufficiently large, significantly smaller than
$1/N^{2}$. It follows that with probability at least $1-1/N$ we have
$|{\hat{A}}(r)-{\hat{P}}(r)|\mathchar 13366\relax N^{-1/2}\log N$ for every
$r$. This implies both the statement about the cardinality of $A$ and the
claim about its uniformity.

For each non-zero $d$ the expectation of the quantity
$\mathop{{{E}}}_{x}A(x)A(x+d)A(x+2d)A(x+3d)$ is equal to
$\mathop{{{E}}}_{x}P(x)P(x+d)P(x+2d)P(x+3d)$, and when $d=0$ it is at most
$1$. Therefore, the expectation of
$\mathop{{{E}}}_{x}A(x)A(x+d)A(x+2d)A(x+3d)$ is at most ${1\over
16}-c+N^{-1}$, where $c$ is the absolute constant obtained in Lemma 5.
Therefore, the probability that $\mathop{{{E}}}_{x}A(x)A(x+d)A(x+2d)A(x+3d)$
is at most ${1\over 16}-{c\over 2}$ is at least $8c$. Since this is bigger
than $N^{-1}$ when $N$ is sufficiently large, we can find a choice of $A$ that
has all the properties claimed. $\mathchar 1027\relax$

The main result of this paper answers the conjecture in [3] but it does not
answer every question one might wish to ask. In particular, it is noticeable
that the number of progressions of length 4 in $A$ was not much less than it
would be for a random set. The following question, which was asked by Ruzsa
and appears in [2] as Problem 3.2, is still open.

###### Problem

Let $A$ be a uniform subset of ${{Z}}_{N}$ of density $\alpha$. Must $A$
contain at least $\alpha^{1000}N^{2}$ arithmetic progressions of length 4?

Of course “1000” is just an informal way of referring to some absolute
constant. This question is open only for progressions of length 4: for longer
progressions, Ruzsa has constructed a counterexample (see Theorem 2.4 of [1]).

References.

[1] V. Bergelson, B. Host. and B. Kra. with an appendix by I. Z. Ruzsa,
Multiple recurrence and nilsequences, Invent. Math., 160 (2005), 261-303.

[2] E. S. Croot and V. F. Lev, Open problems in additive combinatorics, in:
Additive Combinatorics, CRM Proc. Lecture Notes 43, Amer. Math. Soc.
(Providence, RI 2007), pp. 207-233.

[3] W. T. Gowers, A new proof of Szemerédi’s theorem, Geom. Funct. Anal. 11
(2001), 465-588.

[4] K. F. Roth, On certain sets of integers, J. London Math. Soc. 28 (1953),
245-252.

[5] E. Szemerédi, On sets of integers containing no $k$ elements in arithmetic
progression, Acta Arith. 27 (1975), 299-345.

[6] J. Wolf, The minimum number of monochromatic 4-term progressions in
${Z}_{p}$, J. Comb. 1 (2010), 53-68.


# Comprehensive peak-width analysis in matter-wave diffraction under grazing
incidence conditions

Lee Yeong Kim  Department of Physics, Ulsan National Institute of Science and
Technology (UNIST), Ulsan 44919, Republic of Korea  Do Won Kang
<EMAIL_ADDRESS>Department of Chemistry, Ulsan National Institute of
Science and Technology (UNIST), Ulsan 44919, Republic of Korea  Jong Chan Lee
Department of Chemistry, Ulsan National Institute of Science and Technology
(UNIST), Ulsan 44919, Republic of Korea  Eunmi Chae  Department of Physics,
Korea University, Seongbuk-gu, Seoul, Republic of Korea  Wieland Schöllkopf
Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195
Berlin, Germany  Bum Suk Zhao (조범석<EMAIL_ADDRESS>Department of Physics,
Ulsan National Institute of Science and Technology (UNIST), Ulsan 44919,
Republic of Korea  Department of Chemistry, Ulsan National Institute of
Science and Technology (UNIST), Ulsan 44919, Republic of Korea

###### Abstract

Thermal energy atom scattering at a surface with grazing incidence conditions
is an innovative method for investigating dispersive atom-surface interactions
with potential application in quantum sensing interferometry. The complete
establishment of this technique would require a detailed peak analysis, which
has yet to be achieved. We examined peak-width fluctuations in atomic and
molecular beams diffracted by a grating under grazing incidence conditions.
Careful measurements and analyses of the diffraction patterns of He atoms and
D2 molecules from three square-wave gratings with different periods and radii
of curvature enabled the identification of factors influencing the variations
in the width as a function of the grazing incidence angle. The effects of
macroscopic surface curvature, grating magnification, and beam emergence are
substantial under these conditions but negligible for incidence angles close
to the normal. Our results shed light on the phenomena occurring in grazing
incidence thermal energy atom scattering.

X-ray, electron, neutron, and atom diffraction techniques are well-established
methods for studying the crystal structures of materials and their changes
over time. In diffraction experiments, the width of the diffraction peak,
along with its intensity and position, is a critical parameter for sample
analysis. For example, peak-width analysis has been employed to estimate
crystallite or grain sizes and crystal strains in X-ray powder diffraction [1,
2, 3, 4] and in grazing incidence X-ray scattering [5, 6]. In thermal-energy
atom scattering (TEAS), the broadening of peak widths provides insights into
temperature-induced alterations in surface morphology [7] and the density of
defects such as steps on a crystal surface [8, 9].

The design of optical elements such as mirrors and gratings for X-ray and
matter-wave optics also necessitates a comprehensive investigation of the
widths of scattering peaks. X-rays have been focused efficiently using
cylindrical concave mirrors by minimizing peak broadening effects [10, 11,
12]. Recently, this endeavor has also been extended to He atoms [13].
Furthermore, understanding wavelength-dependent peak width broadening is
essential for atom monochromators [14]. Thus, analyzing peak widths is crucial
for developing new methodologies and technologies based on wave diffraction.

Grazing incidence thermal energy atom scattering (GITEAS) at a surface offers
a unique approach that can complement conventional TEAS, akin to the
relationship between X-ray scattering and grazing incidence X-ray scattering.
The lower effective energy and longer wavelength toward the surface for GITEAS
make it more sensitive to weak interactions and less responsive to surface
roughness. As a result, GITEAS has become valuable for studying the dispersive
interaction of atoms with a surface [15, 16, 17]. Furthermore, microstructure-
grating interferometry with GITEAS can be applied for quantum sensing in
conjunction with monolithic atom interferometry using TEAS [18].

The versatile applications of GITEAS necessitate a precise peak-width
analysis. However, under grazing incidence conditions, the peak widths are
strongly influenced by an infinitesimal curvature (curvature radius of a few
kilometers) and the diffraction direction near a surface, which results in
unusual variations in peak widths. The presence of abnormally wide or narrow
peaks further complicates the analysis. Furthermore, the traditional peak-
width analysis scheme used in TEAS is insufficient for GITEAS, which
highlights the need for a more sophisticated approach.

In this letter, we report a comprehensive analysis of peak widths for GITEAS.
By adjusting the grating and incident beam properties, we investigate various
factors contributing to peak-width variations, such as the macroscopic surface
curvature, grating magnification, incident beam divergence, and angular
dispersion. He atoms (D2 molecules) with mean de Broglie wavelengths $\lambda$
of 330 or 140 pm (140 pm) are diffracted at grazing incidence angles up to 30
mrad by three gratings of different periods and macroscopic curvatures. By
comparing the measured peak widths with calculated widths, we identify the
dominant factors influencing the variations in peak width. This resolves any
potential ambiguities in the data analysis caused by extraordinary peak
widths.

Figure 1: Measured angular spectra of matter-wave beams of He (a-d) and D2 (e)
diffracted by grating G1 (a), G20 (b), and G400 (c-e) at incidence angles
$\theta_{\rm in}$ of (a) 5.77, (b) 4.41, (c) 1.29, (d) 0.89, and (e) 0.88
mrad. The horizontal bar near the specular peak indicates the corresponding
width $w_{\rm in}$ of the beam with no grating installed. In (a) and (b), the
1st-order peak is magnified for clarity.

Our atom optics apparatus characterized by a tightly collimated incident beam
and high-angular-resolution detector enables precise measurements of peak
widths. Further details of the setup can be found in previous references [16,
19] and the Supplemental Material [20]. Here, we focus on the aspects of the
apparatus pertinent to the data analysis presented in this study. A continuous
beam of He or D2 is formed by supersonic expansion of the corresponding pure
gas from a source cell. The source temperature ($T_{0}$) and pressure
($P_{0}$) influence the particle velocity distributions. We use three sets of
source conditions, viz., gas species, $T_{0}$, and $P_{0}$: (He, 9.0 K, 0.5
bar), (He, 52 K, 26 bar), and (D2, 52 K, 2 bar). For each set of conditions,
we observe a velocity distribution, from which the mean velocity ($v$), full
width at half maximum (FWHM) ($\Delta v$), and corresponding mean de Broglie
wavelength $\lambda$ (see Fig. SM2 [20]) are determined. Accordingly, we
obtain three corresponding sets of incident beam properties, including $v$,
$\Delta v$, and $\lambda$: (304 m/s, 2.3 m/s, 330 pm), (733 m/s, 5.9 m/s, 140
pm), and (736 m/s, 79 m/s, 140 pm), respectively, which are used to explore
the effects of $\lambda$ and $\Delta v$ separately.

The beam is collimated using two slits (S1 and S2) positioned 100 cm apart as
shown in Fig. SM1 [20]. The widths of these slits, $W_{\rm S1}$ and $W_{\rm
S2}$, are 20 $\mu$m, except for one set of data where $W_{\rm S1}=10$ $\mu$m
and $W_{\rm S2}=15$ $\mu$m. The incident and scattered beams are detected by
precisely rotating a mass spectrometer with electron-bombardment ionization.
The rotational axis of the detector is located 40 cm downstream from S2. A
third slit (S3) with a width of $W_{\rm S3}=25$ $\mu$m is positioned just
before the detector. The distance between the rotational axis and S3, referred
to as the grating–detector distance, is $L=38$ cm.

We employ three square-wave gratings with varying periods $d$ and strip widths
$a$: G1 with $d=1$ $\mu$m and $a=0.25$ $\mu$m; G20 with $d=20$ $\mu$m and
$a=10$ $\mu$m; and G400 with $d=400$ $\mu$m and $a=200$ $\mu$m. Albeit
nominally plane, the grating surfaces exhibit small circular curvatures. Under
grazing incidence conditions the curvature along the direction perpendicular
to the incident plane affects the peak width only negligibly [21, 22].
Therefore, we consider the gratings as cylindrical with their curvature radii
$R$. For concave and convex gratings, $R>0$ and $R<0$, respectively. The
estimated $R$ values of G1, G20, and G400 are 30, $-210$, and 1800 m,
respectively.

Figure 1 shows angular spectra for the three gratings measured at different
experimental conditions for various incidence angles. The graphs represent the
He+ or D${}_{2}^{+}$ signal as a function of the detection angle $\theta$.
Here, $\theta_{\rm in}$ and $\theta$ are measured with respect to the grating
surface. Numbers $n$ indicate the diffraction order assigned to each peak. The
peak positions $\theta_{n}$ and FWHM values $w_{n}$ of the $n$th-order
diffraction beams are determined by fitting each peak to a single Gaussian
function. Similarly, we determine the FWHM $w_{\rm in}$ of the incident beam
spectrum when the grating is removed from the beam path.

A peak-width analysis reveales unexpected irregular hierarchies of $w_{n}$ for
the 5 cases of Fig 1: (a) $w_{0}>w_{1}>w_{\rm in}$, (b) $w_{1}>w_{0}>w_{\rm
in}$, (c) $w_{-1}>w_{\rm in}\approx w_{1}>w_{0}$, (d) $w_{\rm
in}>w_{1}>w_{0}>w_{-1}$, and (e) $w_{-1}>w_{\rm in}>w_{1}>w_{0}$. Generally,
in TEAS, diffraction peak widths increase with $|n|$ owing to the angular
dispersion [23, 24], and an increase of the specular width $w_{0}$ with
respect to $w_{\rm in}$ can be attributed to surface defects [25]. Therefore,
the unexpected hierarchies could lead to misinterpretation of underlying
physics and errors in peak assignment.

Figure 2: Angular FWHM of the $n$th-order diffraction peak, $w_{n}$, as a
function of the incidence angle $\theta_{\rm in}$. The experimental conditions
are the same as those in Fig. 1. The vertical dashed line denotes the Rayleigh
incidence angle of the $-1$st-order diffraction beam, referred to as
$\theta_{{\rm R},-1}$, where the $-1$st-order diffraction beam emerges from
the grating surface. The horizontal dotted line represents the width of the
incident beam, $w_{\rm in}$. In (b), filled symbols within the range of
$\theta_{\rm in}$ from 0.5 to 5.5 mrad illustrate data obtained with tight
collimation, resulting in a narrower $w_{\rm in}$ of 0.081 mrad (thick
horizontal line). The insets in (c)–(e) show seven-fold magnifications of the
corresponding data series.

To study the peak-width variations systematically, we plot $w_{n}$ as a
function of $\theta_{\rm in}$ for the five experimental conditions in Fig. 2.
Each graph includes horizontal dotted lines indicating $w_{\rm in}$ and
vertical dashed lines representing the Rayleigh incidence angle of $-1$st-
order diffraction beam emergence ($\theta_{{\rm R},-1}$). When $\theta_{\rm
in}=\theta_{{\rm R},-1}$, the $-1$st-order diffraction beam emerges from the
grating and propagates parallel to its surface; in this case, $\theta_{-1}=0$
[26, 27].

The relationship between $w_{n}$ and $\theta_{\rm in}$ varies under different
experimental conditions and for individual diffraction orders. Furthermore,
the inconsistent hierarchies among $w_{0}$, $w_{1}$, and $w_{-1}$ change with
$\theta_{\rm in}$. Several factors contribute to these variations: (i) the
macroscopic curvature of the grating surface, (ii) grating magnification,
(iii) diffraction beam spread resulting from the divergence of the incident
beam, and (iv) angular dispersion due to the non-monochromatic nature of the
beam. Among these factors, (i) pertains to a property of the grating, (ii)
results from the diffraction principle, and (iii) and (iv) are determined by
the incident beam properties.

The macroscopic curvature of the grating surface accounts for the variation in
$w_{0}$ shown in Fig. 2. The magnitude of $|R|$ directly influences the
steepness of the decrease in $w_{0}$. Additionally, when $R>0$ ($R<0$),
$w_{0}$ increases (decreases) asymptotically toward $w_{\rm in}$ with
$\theta_{\rm in}$ as illustrated in Fig. 2(a) [Fig. 2(b)]. Furthermore, the
steep increase in $w_{-1}$ with decreasing $\theta_{\rm in}$ in Fig. 2(a) and
2(e) is attributed to angular dispersion. Finally, the hierarchical order of
$w_{1}>w_{0}>w_{-1}$ shown in the inset of Fig. 2(d) results from the grating
magnification.

Factors (i)–(iv) contribute to $w_{n}$ differently depending on $n$,
$\lambda$, and $\theta_{\rm in}$, which can be formulated by the following
approximate equation for a linear width $W_{n}=Lw_{n}$:
$W_{n}=\sqrt{[W_{n}^{(1)}]^{2}+[W_{n}^{(2)}]^{2}+[W_{n}^{(3)}]^{2}+[W^{(4)}_{n}]^{2}+[W^{(5)}]^{2}}$,
where

$W^{(1)}_{n}=\left(\frac{1}{o}+\frac{1}{L}-\frac{1}{f_{n}}\right)LW_{\rm
G}\frac{\sin\theta_{n}}{\sin\theta_{\rm in}}\,,$ (1)
$W^{(2)}_{n}=\frac{2.355}{4}\frac{W_{\rm S1}}{o}L\frac{\sin\theta_{\rm
in}}{\sin\theta_{n}}\,,$ (2) $W^{(3)}_{n}=0.884\frac{\lambda}{W_{\rm
S2}}L\frac{\sin\theta_{\rm in}}{\sin\theta_{n}}\,,$ (3)
$W^{(4)}_{n}=\left(\frac{|n|\lambda}{d}\right)\frac{\Delta
v}{v}\frac{1}{\sin\theta_{n}}L\,,$ (4)

and

$W^{(5)}=\frac{2.355}{4}W_{\rm S3}\,.$ (5)

In these equations, $o$ represents the object distance, $f_{n}$ denotes the
focal length of the $n$th-order diffraction beam, and $W_{\rm G}$ denotes the
width of the incident beam at the center of the grating. Because S1 constrains
the effective source size of the matter-wave beam, we approximate the object
as a Gaussian distribution with a standard deviation of WS1/4. Similarly, the
boxcar-shaped function defined by S3 is approximated as a Gaussian function
with a standard deviation of WS3/4.

The macroscopic curvature of the grating represented by $R$ is relevant to its
focal length. Under grazing incidence conditions, the object distance $o$ and
image distance $i_{n}$ of the $n$th-order beam satisfy the thin lens equation:
$1/o+1/i_{n}=1/f_{n}$. The term in the parentheses of Eq. (1) then represents
the focusing error, $\epsilon_{n}$, and the product $\epsilon_{n}LW_{\rm G}$
is the width of the (de-)focused incident beam at the detection plane [28].
The focal length $f_{n}$ is a function of $R$ and $\theta_{\rm in}$.
Specifically, $f_{0}$ can be expressed as $f_{0}=R\sin\theta_{\rm in}/2$ [10,
13]; therefore, $i_{n}$ (or $\epsilon_{n}$) can be determined based on the
values of $R$ and $\theta_{\rm in}$, as illustrated in Fig. SM4 [20].

The grating magnification given by $M_{n}=\sin\theta_{n}/\sin\theta_{\rm in}$,
also known as anamorphic magnification, represents the ratio of the width of a
collimated diffracted beam to that of a collimated incident beam [29]. When
considering a collimated beam ($o\rightarrow\infty$) incident on a flat
grating ($f_{n}\rightarrow\infty$), $W^{(1)}_{n}$ characterizes the grating
magnification.

Figure 3: Comparison between the measured (symbols) and calculated (solid
curves) linear widths $W_{n}$ of the $n$th-order diffraction beam. The first-
and second-column data refer to G1 and G20, respectively. The data are listed
sequentially from the top for $n=0$, 1, and $-1$. $W_{0}$, $W_{1}$, and
$W_{-1}$ are plotted separately with the calculated $W^{(1)}_{n}$,
$W^{(2)}_{n}$, $W^{(3)}_{n}$, $W^{(4)}_{n}$, and $W^{(5)}$.

$W^{(2)}_{n}$ describes the effect of the geometrical incidence beam
divergence $\simeq W_{\rm S1}/o$ on the diffraction beam spread
$\Delta\theta_{n}$. When $W_{\rm S1}=20$ $\mu$m, $W^{(2)}_{n}=3.2$ $\mu$m for
the specular peak, which is negligible. In contrast, $W^{(2)}_{n}$ becomes
significant when $\theta_{n<0}\rightarrow 0$ as it happens when $\theta_{\rm
in}$ approaches the Rayleigh angle of beam emergence $\theta_{{\rm R},n}$.

Slit diffraction at S2 contributes additional incident-beam divergence
$\Delta\theta_{\rm in,SD}$, which is responsible for $W^{(3)}_{n}$. In the
Fraunhofer limit, $\Delta\theta_{\rm in,SD}=0.844\lambda/W_{\rm S2}$. Similar
to $W^{(2)}_{n}$, this contribution becomes pronounced only for an emerging
peak close to the Rayleigh condition.

$W^{(4)}_{n}$ accounts for diffraction peak broadening resulting from angular
dispersion due to the beam’s finite velocity spread $\Delta v$. This effect is
absent in the specular peak.

Finally, the finite size of the detector entrance slit, S3, also contributes
to the observed diffraction peak widths by the term ($W^{(5)}$) of Eq. (5).
This contribution is a constant 15 $\mu$m for all experimental conditions.

To assess the relative contributions of these five terms to $W_{n}$, we
compare the measured $W_{n}$ (symbols) for G1 and G20 with the corresponding
calculated values for $W_{n}$, $W^{(1)}_{n}$, $W^{(2)}_{n}$, $W^{(3)}_{n}$,
$W^{(4)}_{n}$, and $W^{(5)}$ (lines) in Fig. 3. The theoretical curves are
determined considering the grating’s macroscopic curvature radius $R$ as the
sole adjustable parameter. The angular width $w_{n}$ presented in Figs. 2(a)
and 2(b) is converted to the linear width $W_{n}$ and presented in the first-
and second-column graphs in Fig. 3, respectively. The values for $W^{(1)}_{n}$
corresponding to G1 (G20) are calculated with $W_{\rm G}=38.5$ $\mu$m ($W_{\rm
G}=34.2$ $\mu$m). The estimation of $W_{\rm G}$ employs the linear width of
the incident beam, namely, $W_{\rm in}=51.2$ $\mu$m ($W_{\rm in}=46.0$
$\mu$m).

For G20 a second data set was obtained with tighter beam collimation ($W_{\rm
S1}=10$ $\mu$m, $W_{\rm S2}=15$ $\mu$m) at small incidence angles $\theta_{\rm
in}<5.5$ mrad (filled symbols in Fig. 2(b)) resulting in an angular width as
small as $w_{\rm in}$ = 0.081 mrad ($W_{\rm in}=30.8$ $\mu$m and $W_{\rm
G}=21.2$ $\mu$m). Figures 3(d)–3(f) illustrate the corresponding calculations.

The breakdown of $W_{n}$ into its five constituent terms in Fig. 3 highlights
the dominant factors in each case. The substantial reductions in $W_{n}$ for
G1 at varying incidence angles result from different factors for $n=0$, 1, and
$-1$. As illustrated in the left-column graphs, at $\theta_{\rm in}<10$ mrad,
$\theta_{\rm in}<5$ mrad, and $\theta_{\rm in}<30$ mrad, $W^{(1)}_{0}$,
$W^{(1)}_{1}$ (red dashed curves), and $W^{(4)}_{-1}$ (blue-dotted curve)
predominantly influence their respective $W_{n}$ values. Notably, the grating
magnification is unity for the specular beam, which makes $\epsilon_{0}$ the
key determinant for $W^{(1)}_{0}$. Conversely, $\epsilon_{1}$ varies by less
than 33$\%$ in the given range of incidence angles while the grating
magnification term $M_{1}$ decreases tenfold [see Figs. SM4(c1) and SM4(d1)]
[20]. Therefore, the principal contributors to the steep decline in $W_{n}$
for $n=0$, 1, and $-1$ are the macroscopic curvature, grating magnification,
and angular dispersion, respectively.

Similar to G1, the decreases in $W_{0}$ and $W_{1}$ for G20 are primarily
determined by the macroscopic curvature and grating magnification,
respectively [see Figs. 3(d), 3(e), SM4(c2), and SM4(d2)]. However, unlike G1,
the 20-fold larger $d$ of G20 diminishes the effect of angular dispersion [see
Eq. (4)]; as a result, the influence of curvature on the reduction of $W_{-1}$
becomes dominant.

Even though G400 is nearly flat with $R=1800$ m, given the extreme grazing-
incidence conditions the curvature still affects the peak width variations. As
can be seen in Figs. 2(c)–2(e) all three peak widths $w_{1}$, $w_{0}$, and
$w_{-1}$ can be narrower than the incidence beam width $w_{\rm in}$ resulting
from beam focusing due to the concave curvature of the grating (refer to the
first and second rows in Fig. SM5) [20].

In addition, the width hierarchy visible in Fig. 2(d) ($w_{1}>w_{0}>w_{-1}$)
presents a clear example for peak widths being dominated by grating
magnification; the closer a beam propagates to the surface, the smaller its
width. This trend is less clear in Fig. 2(c) and (e) where $w_{-1}$ is not
consistently the smallest width at the given incidence angles. Specifically,
for incidence angles slightly larger than the $-1$-order Raleigh angle
$\theta_{{\rm R},-1}$, $w_{-1}$ exceeds both $w_{0}$ and $w_{1}$.

As shown in Figs. SM5(c), SM5(f), and SM5(i), the larger contributions of the
terms $W^{(2)}_{-1}$, $W^{(3)}_{-1}$, and $W^{(4)}_{-1}$ compared with
$W^{(1)}_{-1}$ lead to broadening of $W_{-1}$. The small $\theta_{-1}$ close
to Rayleigh conditions boosts these three terms. Interestingly, as shown in
Fig. SM5, $W^{(2)}_{-1}$ and $W^{(3)}_{-1}$ were the dominant factors for the
He atom beams with two different $\lambda$, whereas $W^{(4)}_{-1}$ was the
crucial factor for the D2 molecular beam. This behavior can be attributed to
the 13-times larger velocity spread $\Delta v$ of the D2 beam compared with
that of the He beam at identical velocity $v$.

In conclusion, our combined experimental and theoretical investigations of
diffraction peak widths in GITEAS reveal the primary factors that induce
variations in peak widths. Notably, the primary factor governing the width of
diffraction beams varies depending on diffraction order, incidence angle, and
grating period. Our study reveals the effects of macroscopic curvature,
emerging beams, and grating magnification, which have been overlooked in other
scattering techniques, such as TEAS. Thus, our findings address potential
ambiguities in interpreting diffraction data as those presented in Fig. 1.

The comprehensive peak-width analysis conducted in this study lays the
groundwork for extending the applicability of GITEAS to investigate the unique
characteristics of dispersive interactions between atoms and thin-layer
surfaces such as graphene sheets or few layers hexagonal boron nitride, known
for their flexibility. While recent theoretical investigations [30] have
delved into these interactions, limited experimental studies are available.
Additionally, this analysis can guide the design of atom optical components.
Although both $w_{1}$ at $\theta_{\rm in}=3$ mrad and $w_{-1}$ near
$\theta_{{\rm R},-1}$ in Fig. 2(a) are sufficiently broad for monochromator
applications, only the $-1$st-order diffraction beam is suitable for this
purpose. This is because wavelength-dependent angular dispersion and
wavelength-independent grating magnification primarily influence $w_{-1}$ and
$w_{1}$, respectively. Furthermore, peak-width analysis will become critical
in atom interferometry using GITEAS.

This study was supported by NRF (National Research Foundation of Korea) grants
funded by the Korean Government (NRF-2020R1A2C3003701 and
NRF-2022M3C1C8094518).

L. Y. K. and D. W. K. contributed equally to this work.

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Supplemental Material

Atomic and molecular beam apparatus

Figure SM1: Schematic of the experimental setup. The distances between the
components are not drawn to scale. The incidence and detection angles,
$\theta_{\rm in}$ and $\theta$, are measured with respect to the grating
surface. The inset illustrates a grating with $d$ and $a$ representing its
period and strip width, respectively.

A continuous beam was generated by supersonically expanding either He atoms or
D2 molecules through a 5-$\mu$m-diameter nozzle. The temperature $T_{0}$ and
pressure $P_{0}$ of the source were set to 9.0 K and 0.5 bar, respectively,
for He atoms of $\lambda=330$ pm and set to $T_{0}=52$ K and $P_{0}=26$ bar,
respectively, for He atoms of $\lambda=140$ pm. To compare the diffractions of
the He atomic and D2 molecular beams of the same $\lambda$ values, we set the
conditions for D2 molecules to $P_{0}=2$ bar and $T_{0}=52$ K. The $\lambda$
values were determined from the velocity distributions of the particles.

The beam first passed through a skimmer with a diameter of 500 $\mu$m and was
subsequently collimated by two slits (S1 and S2) placed 100 cm apart. The
widths of these slits, $W_{\rm S1}$ and $W_{\rm S2}$, were the same ($W_{\rm
S1}=W_{\rm S2}=20$ $\mu$m) for most measurements while for one set of data
shown in this paper it was $W_{\rm S1}=10$ $\mu$m and $W_{\rm S2}=15$ $\mu$m.
The incident and scattered beams were detected by precisely rotating a mass
spectrometer with electron-bombardment ionization. The rotational axis of the
detector was located 40 cm downstream from S2. Just before the ion detector, a
third slit (S3) with a width of $W_{\rm S3}=25$ $\mu$m was positioned to
enhance the angular resolution. The distance between the rotational axis and
S3, referred to as the grating–detector distance (or detector radius), was
$L=38$ cm. This distance, combined with the narrow aperture of S3, yielded a
full width at half maximum (FWHM) of approximately 0.12 mrad for the
collimated incident beam. The grating was carefully positioned and oriented
such that the pivot axis of the detector aligned with the center of the
grating plane and was maximally parallel to the grating grooves. Therefore,
the grating grooves were perpendicular to the plane of incidence defined by
the incident wave vector and grating normal, resulting in in-plane
diffraction. This geometry is illustrated in Fig. 1 in the main text, where
both the incidence angle $\theta_{\rm in}$ and detection angle $\theta$ are
measured in relation to the grating plane.

Figure SM2: Measured velocity distributions of He (red squares) and D2 (blue
circles) at (a) $T_{0}=$ 9 and (b) 52 K. The intensity is scaled to the peak
values. Under conditions of $P_{0}=0.5$ bar and 26 bar for $T_{0}=$ 9 K and 52
K, respectively, the FWHM values for the He atom beams are 2.28 and 5.9 m/s.
In contrast, the velocity width of the D2 molecular beam is 79 m/s at $T_{0}=$
52 K. The mean velocities for these distributions are approximately 304 and
730 m/s for $T_{0}=$ 9 and 52 K, respectively.

Velocity distributions of atomic and molecular beams
We obtained the velocity distributions of the He atomic and D2 molecular beams
using time-of-flight measurements conducted with a mechanical pseudo-random
beam chopper [31]. Figures SM2(a) and SM2(b) present the results obtained at
temperatures of $T_{0}=9.0$ K and 52 K, respectively. We employed Gaussian
function fitting to determine both the mean velocity $v$ and FWHM $\Delta v$.
The ratios of $\Delta v/v$ for the He atom beams are 0.0075 and 0.0081 at the
two different $T_{0}$ values, which are similar. In contrast, the ratio
$\Delta v/v$ for the D2 molecular beam is 0.11.

Fabrication and characterization of the diffraction gratings

Table SM1: Grating properties relevant to the peak-width variation |  | period |  | curvature radius
---|---|---|---|---
G1 |  | 1 $\mu$m |  | 30 m
G20 |  | 20 $\mu$m |  |  $-210$ m
G400 |  | 400 $\mu$m |  | 1800 m

We employed three square-wave gratings with varying periods $d$ and strip
widths $a$: G1 with $d=1$ $\mu$m and $a=0.25$ $\mu$m, G20 with $d=20$ $\mu$m
and $a=10$ $\mu$m, and G400 with $d=400$ $\mu$m and $a=200$ $\mu$m. G1 and G20
are microstructured arrays, each measuring 56 mm in length, consisting of
110-nm-thick chromium stripes that are 5 mm in length and are deposited on a
2-mm-thick quartz surface. In contrast, G400 is an array featuring parallel
photoresist strips with a thickness of 1 $\mu$m, width of 200 $\mu$m, and
length of 4 mm. These strips are located on a commercial gold mirror (Thorlabs
PFSQ20-03-M03), which is 6 mm thick and has a surface area of $50.8\times
50.8$ mm2. Only the stripes were exposed to the incident atomic beam for all
the angles of incidence examined in this study. Table SM1 provides a summary
of the properties of these gratings that are pertinent to variations in peak
width, including their period $d$ and radius of curvature $R$.

Example angular spectrum
Figure SM3 shows the full angular spectrum of the He atom beam diffracted from
G1 at $\theta_{\rm in}=5.77$ mrad [Fig. 1(a)]. The specular peak position
$\theta_{0}$ equals $\theta_{\rm in}=5.77$ mrad. The widths of specular and
1st-order-diffraction peaks ($w_{0}$ and $w_{1}$) are 0.31 mrad and 0.21 mrad,
respectively.

Figure SM3: Angular spectrum of the He atom beam diffracted from G1 at
$\theta_{\rm in}=5.77$ mrad (the insets show parts of the spectrum on an
enlarged scale).

Experimental conditions
Table SM2 lists the five experimental conditions E1–E5 at which the five data
sets presented in this study were obtained.

Table SM2: Experimental conditions employed in this experiment  |  |  | Particle |  | Grating |  | $\lambda_{\rm dB}$ |  | $\textit{T}_{0}$ |  | $\textit{P}_{0}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---
| E1 |  | He |  | G1 |  | 330 pm |  | 9.0 K |  | 0.5 bar |
| E2 |  | He |  | G20 |  | 330 pm |  | 9.0 K |  | 0.5 bar |
| E3 |  | He |  | G400 |  | 330 pm |  | 9.0 K |  | 0.5 bar |
| E4 |  | He |  | G400 |  | 140 pm |  | 52 K |  | 26 bar |
| E5 |  | D2 |  | G400 |  | 140 pm |  | 52 K |  | 2.0 bar |
Figure SM4: (a1–a5) Image distance $i_{n}$, (b1–b5) focal length $f_{n}$,
(c1–c5) focusing error $\epsilon_{n}$, and grating magnification $M_{n}$ of
the $n$th-order diffraction beam. The five columns correspond to the
experimental conditions outlined in Table SM2, respectively. Here, G1, G20,
and G400 are assumed to be cylindrical mirrors with curvature radii of $R=30$,
$-210$, and 1800 m, respectively. These graphs are plotted as functions of the
incidence angle $\theta_{\rm in}$. The horizontal dotted line in “(a1–a5)”
indicates the grating–detector distance $L=0.38$ m. Figure SM5: Comparison
between the measured (symbols) and calculated (solid curves) linear widths
$W_{n}$ of the $n$th-order diffraction beam from G400. The graphs in the three
columns correspond to the experimental conditions E3, E4, and E5 as listed in
Table SM2, respectively. The data are presented sequentially from the top for
$n=0$, 1, and $-1$. Specifically, $W_{0}$, $W_{1}$, and $W_{-1}$ are
individually plotted alongside the calculated values $W^{(1)}_{n}$,
$W^{(2)}_{n}$, $W^{(3)}_{n}$, $W^{(4)}_{n}$, and $W^{(5)}$.

Calculation of parameters required for peak-width analysis
For a cylindrically concave mirror with a curvature radius $R$, the object
distance $o$ and image distance $i_{n}$ of the $n$th-order beam satisfy the
following equation under grazing incidence conditions:

$\frac{\theta_{\rm
in}^{+}-\alpha/2}{o-R\alpha}+\frac{\theta_{n}^{+}+\alpha/2}{i_{n}+R\alpha}=\frac{2}{R}+\frac{(\theta_{\rm
in}^{-}-\theta_{n}^{-})+(\theta_{\rm in}^{+}-\theta_{n}^{+})}{W_{\rm
G}/\theta_{\rm in}}\,.$ (SM1)

Here, $\theta_{\rm in}^{+}$ and $\theta_{\rm in}^{-}$ represent the outermost
values of the incidence angles that result in $\theta_{n}^{+}$ and
$\theta_{n}^{-}$, respectively. The incidence beam spreads over a distance of
$W_{\rm G}/\sin\theta_{\rm in}$ on the surface. This chord subtends an angle
of $2\alpha$ at the center of curvature, making $\alpha$ approximately equal
to $W_{\rm G}/(2R\sin\theta_{\rm in})$. Consequently, $i_{n}$ varies as a
function of $\theta_{\rm in}$, as depicted in the first row of Fig. SM4 for
the five experimental conditions. For our calculations, we assume $R=30$ m.
When $i_{n}=L$, a diffraction beam is focused on the detection plane. Eq. SM1
transforms into the thin-lens equation, $1/o+1/i_{0}=1/f_{0}$, for a specular
beam of $n=0$ with $f_{0}=R\sin\theta_{\rm in}/2\simeq R\theta_{\rm in}/2$
under grazing incidence conditions. Generally, $1/o+1/i_{n}=1/f_{n}$, with
which we obtain $f_{n}$ using $i_{n}$ [See Figs. SM4(b1–b5)]. Then,
$\epsilon_{n}=1/L-i_{n}$. The third row of Fig. SM4 shows $|\epsilon_{n}|$ for
the five experimental conditions. To elucidate the contributions of the
focusing error $\epsilon_{n}$ and the grating magnification $M_{n}$ to
$W_{n}^{(1)}$, we plot $M_{n}$ in the last row of Fig. SM4.

Peak-width analysis for G400
Similar to Fig. 3 in the main text, Fig. SM5 compares the measured $W_{n}$
(symbols) for G400 with the corresponding calculated values: $W_{n}$,
$W^{(1)}_{n}$, $W^{(2)}_{n}$, $W^{(3)}_{n}$, $W^{(4)}_{n}$, and $W^{(5)}$
(solid lines). The theoretical curves are derived using $R=1800$ m. In most
cases, the primary factor is $W^{(1)}_{n}$ (red dashed line). Exceptions occur
for $W_{-1}$ at the Rayleigh conditions of $\theta_{\rm in}=\theta_{\rm
R,-1}$. At these conditions, $\theta_{-1}$ approaches 0, leading to large
values for $W^{(2)}_{-1}$, $W^{(3)}_{-1}$, and $W^{(4)}_{-1}$.


where $\mathds{C}$ is, in principle, an arbitrary point which however we choose to be equal to $-\infty^+$.

We stress that the fermion in the external field differs from the gauge-invariant fermion field which appears
in the string operator, see Eqn. (<ref>).
Indeed, as well-known (see, for example, [72, 73]) in order to get the
gauge-invariant string operator it is necessary to include the gauge boson (gluon) radiations from the fermions
after the interaction of them with photons (or other gauge bosons) as shown in Fig. <ref>, the left panel.
Otherwise, we deal with the fermions in the external fields which are not gauge-invariant
(see, Fig. <ref>, the right panel).

To illustrate the last statement, let us consider the simplest case of Compton-like amplitude (see also [73]).
We have

\begin{eqnarray}
\label{CA-1}
T^{\mu\nu}=\int (d^4x) e^{-iq\cdot x} \langle p | T J^{\mu}(x) J^{\nu}(0)|p \rangle.
\end{eqnarray}

On the handbag diagram level, we have

\begin{eqnarray}
\label{CA-2}
&&T^{\mu\nu}=\int (d^4x) e^{-iq\cdot x}\times
\\
&&\langle p |
: \bar\psi(x)\gamma^{\mu}
\contraction{}{\psi}{(x)}{\bar\psi} \psi(x) \bar\psi(0)
\gamma^{\nu}\psi(0) : |p \rangle.
\nonumber
\end{eqnarray}

In order to include all gauge boson radiations from the fermion propagator given
by the fermion contraction, we merely make a substitution
(modulo the conventional normalizations which are now irrelevant)

\begin{eqnarray}
\label{SubProp-1}
\contraction{}{\psi}{(x)}{\bar\psi} \psi(x) \bar\psi(0) = S^c(x)\Longrightarrow
\mathrm{S}(x,0)
\end{eqnarray}

with $\mathrm{S}(x,0)$ being gauge-noninvariant Green function, see Eqn. (<ref>).
Using the relation which is similar to Eqn. (<ref>), we can obtain that

\begin{eqnarray}
\label{CA-3}
&&T^{\mu\nu}=\int (d^4x) e^{-iq\cdot x}\times
\\
&&\langle p |
: \bar\psi(x)\gamma^{\mu}
S^{c}(x)\,[x \,;\, 0]_{A}
\gamma^{\nu}\psi(0) : |p \rangle.
\nonumber
\end{eqnarray}

After the factorization procedure, the matrix combination
$\gamma^{\mu}\,S^{c}\,\gamma^{\nu}$ refers to the so-called hard part,
while the non-perturbative hadron matrix element involves the gauge-invariant
string operator defined as

\begin{eqnarray}
\langle p |
: \bar\psi(x)\,[x \,;\, 0]_{A} \psi(0) : |p \rangle.
\end{eqnarray}

§ LOCAL AND NONLOCAL GAUGE TRANSFORM CONVENTION

Before going further, it is important to remind the convention system regarding the gauge transformations which match
the corresponding Wilson path functional (see [55] for more details).
In what follows, for the sake of shortness, we say simply the Wilson line independently on the form of a path unless it
leads to misunderstanding.
For the non-Abelian gauge theory, let us now assume that the fermion and gauge fields are transformed as

\begin{eqnarray}
\label{Wl-GT-1}
&&\psi^{\theta}(x)=e^{+i\theta(x)} \psi(x)\equiv U(x) \psi(x) ,
\\
\label{Wl-GT-1-2}
&&A_{\mu}^\theta(x)= U(x)A_\mu(x) U^{-1}(x) + \frac{i}{g} U(x)\partial_\mu U^{-1}(x),
\end{eqnarray}

where $\theta=\theta^a T^a$ with $T^a$ being the generators of corresponding representations.
With the local transformations fixed as in Eqns. (<ref>) and (<ref>),
we can readily see that the covariant derivative and
the gauge-invariant fermion string operator take the following forms:

\begin{eqnarray}
\label{Wl-GT-2}
i\, {\cal D}_{\mu} = i\, \partial_\mu + g A_\mu(x),
\quad
\mathbb{O}^{\text{g.-inv.}}(x,y)=\bar\psi(y) [y \,;\, x]_{A} \psi(x)
\end{eqnarray}

with the Wilson line defined as

\begin{eqnarray}
\label{WL-def-1}
[x\, ; \, x_0 ]_A =
%\mathbb{P}\text{exp} \Big\{ ig\int_{x_0}^{x} d\omega_\mu A_\mu(\omega)\Big\}
\mathbb{P}\text{exp} \Big\{ ig\int_{P(x_0,x)} d\omega_\mu A_\mu(\omega)\Big\}
= {\bf g}(x | A)\equiv {\bf g}(P),
\end{eqnarray}

where $P(x_0,x)$ stands for a path connecting the starting $x_0$
and destination $x$ points in the Minkowski space.

Inserting the point $x_0$ in the Wilson line of the gauge-invariant string operator, see Eqn. (<ref>), we get that

\begin{eqnarray}
\label{WI-GT-3}
\mathbb{O}^{\text{g.-inv.}}(x,y)=\bar\psi(y) [y \,;\, x_0]_{A} [x_0 \,;\, x]_{A} \psi(x).
\end{eqnarray}

Eqn. (<ref>) hints that the path-dependent non-local gauge transformation of fermions can be introduced in the form of

\begin{eqnarray}
\label{Wl-GT-4}
\psi^{{\bf g}}(x)  =  {\bf g}^{-1}(x | A) \psi(x),
\end{eqnarray}

where $\psi^{{\bf g}}(x)$ is nothing but the Mandelstam gauge-invariant
fermion field $\Psi(x | A)$, modulo the global gauge transforms [54, 74].
This transformation leads, in turn, to (cf. Eqn. (<ref>))

\begin{eqnarray}
\label{A-trans-CG}
A_{\mu}^{{\bf g}}(x)= {\bf g}^{-1}(x | A)A_\mu(x) {\bf g}(x | A) + \frac{i}{g} {\bf g}^{-1}(x | A)\partial_\mu {\bf g}(x | A).
\end{eqnarray}

Therefore, we have the following correspondence between local and non-local gauge transformation

\begin{eqnarray}
\label{Corr-loc-nonloc}
U(x) \Leftrightarrow {\bf g}^{-1}(x | A)
\end{eqnarray}

which is extremely important for the further discussions because the wrong correspondence results in the
substantially wrong conclusions (see for example [75]).

§ GAUGE AND RESIDUAL GAUGE SYMMETRIES

In this Appendix, we remind some subtleties related to the residual gauge transformations
in different gauge theories.

§.§ Classical $U(1)$-gauge theory (Abelian theory)

The $U(1)$-gauge theory where the gauge transformation

\begin{eqnarray}
\label{U1}
A_\mu^\Lambda(x)=A_\mu(x) + \partial_\mu \Lambda(x),
\end{eqnarray}

defines an orbit on the $U(1)$-group.
In the Abelian case, the strength  tensor $F_{\mu\nu}$ is gauge-invariant and, therefore,
only the longitudinal (unphysical) components of field, $A^L_\mu$, can be gauge-transformed.
Indeed, in the classical gauge theory for both $k^2=0$ and $k^2\neq 0$,
the solution of the Maxwell equation in vacuum, $\partial_\mu F_{\mu\nu}=0$, reads (modulo the complex conjugated terms),
(see, e.g., [76])

\begin{eqnarray}
\label{SolMeq}
&&A_{\mu}(x)=A^L_{\mu}(x) + A^\perp_{\mu}(x)=
\\
&&\int(d^4k)e^{ikx} k_\mu a_L(k) +
\int(d^4k)e^{ikx} \delta(k^2) e^{\perp\,(\alpha)}_\mu a_\perp^{(\alpha)}(k),
\nonumber
\end{eqnarray}

where $(d^4 k)$ stands for the corresponding integration measure with an appropriate normalization,
$\alpha=(1,2)$ and $k\cdot e^{\perp\,(\alpha)}=0$. With this expression, we can easily derive the gauge transformations in
$p$-space ($=$ the momentum representation)

\begin{eqnarray}
\label{GTr-pspace-1}
k_\mu a_{L}^{\Lambda}(k) + e^{\perp}_\mu a_{\perp}^{\Lambda}(k) =
k_\mu a_{L}(k) + e^{\perp}_\mu a_{\perp}(k) + k_\mu\tilde\Lambda(k),
\end{eqnarray}

where the imaginary factor $i$ is absorbed in the definition of $\tilde\Lambda$.
In what follows summation over $\alpha$ and dimensionful normalizations are not shown explicitly
unless it leads to misunderstanding.

Since $k\cdot e^{\perp\,(\alpha)}=0$, we conclude that

\begin{eqnarray}
\label{GTr-pspace-2}
a_{L}^{\Lambda}(k)= a_{L}(k) + \tilde\Lambda(k), \quad a_{\perp}^{\Lambda}(k) = a_{\perp}(k),
\end{eqnarray}

or, equivalently,

\begin{eqnarray}
\label{GTr-pspace-3}
&&A^{L,\,\Lambda}_\mu(k)= A^{L}_\mu(k) + k_\mu \tilde\Lambda(k),
\nonumber\\
&&A^{\perp,\,\Lambda}_\mu(k) = A^{\perp}_\mu(k).
\end{eqnarray}

Moreover, it is easy to demonstrate that

\begin{eqnarray}
\label{AL}
A^{L}_\mu(x) = -i \partial_\mu\alpha(x),
\end{eqnarray}

where $\alpha(x)$ is a scalar function which is related to $a_{L}(k)$ via the Fourier transformation,
$\alpha(x)\stackrel{\text{F}}{=}a_{L}(k)$, and
$a_{L}(k)=\xi(k)/k^2$ with $\xi(k)\stackrel{\text{def}}{=}k\cdot A^{L}(k)\neq 0$
for $k^2\neq 0$. Notice that if $k^2=0$, the Maxwell equation takes the simplest form, $k\cdot A(k)=0$,
in the $p$-space and, therefore, $k\cdot A^{L}(k)= k^2 a_{L}(k)=0$ or, in other words,
$a_{L}(k)=\xi(k)/k^2\sim 0/0$.

As well-known, to fix the certain  representative on the group orbit we have to impose a
gauge condition $F(A^\Lambda)=0$ on the gauge-transformed fields in order to find a solution with respect to
the gauge parameter $\Lambda$. Here, we do not discuss the appearance of Gribov's ambiguity.

The Lorentz gauge. As the first example, we consider the Lorentz (covariant) condition which states

\begin{eqnarray}
\label{L-con}
\partial_\mu A^\Lambda_\mu(x)= \partial_\mu A_\mu + \partial^2 \Lambda(x)
\end{eqnarray}

In $p$-space, the condition (<ref>) takes the following form:

\begin{eqnarray}
\label{L-con-2}
k_\mu A^L_\mu(k) + k^2 \tilde\Lambda(k)=0
\end{eqnarray}

which gives us the relation $a_L(k)=-\tilde\Lambda(k)$ for the case of $k^2\neq 0$.
Notice that if $k^2=0$, then the functions $a_L(k)$ and $\tilde\Lambda(k)$ in the
combination $a_L(k)+\tilde\Lambda(k)$ are free functions and they are independent of each other.

However, the gauge condition (<ref>) (or (<ref>)) can not fix the orbit
representative uniquely. Indeed, there is still the so-called residual gauge freedom
defined by $F(A^\Lambda)=F(A)=0$. For the Lorentz condition,
two simultaneous conditions:

\begin{eqnarray}
\label{ResG-1}
\partial_\mu A^\Lambda_\mu(x)=0\quad\text{and}\quad \partial_\mu A_\mu(x)=0
\end{eqnarray}

lead to

\begin{eqnarray}
\label{ResG-2}
\partial^2 \Lambda_0(x)=0
\end{eqnarray}

where the gauge function (parameter) $\Lambda_0$ defines the residual gauge freedom.
That is, the residual gauge transformation with the function $\Lambda_0$ keeps the gauge
condition, $F(A)=0$, gauge-invariant.  Hence, the gauge freedom fixing
means that one fixes all gauge freedom including the residual gauge.
In other words, if there is no residual gauge transformation, the
given gauge condition fixes the gauge freedom completely and we deal with
one representative on a gauge orbit.

Let us consider the second gauge condition in Eqn.  (<ref>). In $p$-space, it leads to
the following possibilities ($k\cdot a_\perp=0$ by definiton)

\begin{eqnarray}
\label{L-con-p-1}
k^2 a_L(k)=0\,\, \Longrightarrow
\begin{cases}
k^2=0, &a_L(\vec{k})-\,\text{arbitrary }\,\\
k^2\neq 0, &a_L(k)=0.\\
\end{cases}
\end{eqnarray}

Hence, we can see that the gauge condition (<ref>) cannot eliminate the
unphysical field $A^L_\mu$ for the case of $k^2=0$. Working with the equation
(<ref>), in the same manner, we conclude that the gauge function
$\tilde\Lambda_0(\vec{k})$ is not fixed and generates the residual gauge transformation provided

It is instructive to consider the condition (<ref>) in the coordinate representation ($x$-space).
Solutions (<ref>) can be easily found and represented, for instance, the following form:

\begin{eqnarray}
\label{Sol-ReG-x}
\Lambda_0(x)=\begin{cases}
\text{const} &\,\\
1/x^2, &\text{for}\,\,\, x^2\neq 0\,\\
C_0 e^{i(x_0-\vec{x}\vec{N})}&\text{with},\,\,\, |\vec{N}|=1.\\
\end{cases}
\end{eqnarray}

Notice that the scalar function $\alpha(x)$ in Eqn. (<ref>) which obeys the second condition in Eqn.  (<ref>),
i.e. $\partial^2\alpha(x)=0$,  has formally the same form as (<ref>).

For $k^2\neq0$, the scalar gauge function $\Lambda$ gives also
the longitudinal (unphysical) field $A^L_\mu$, see (<ref>). Therefore,
the first two solutions of (<ref>) are irrelevant for our study.
In order to get matched with the corresponding condition (<ref>) in the momentum representation,
we have to put $C_0$ be equal to zero, $C_0=0$. However, for the case of $k^2=0$,
as above-mentioned, the functions $\alpha(x)$ and $\Lambda_0(x)$ are independent and arbitrary
due to the different free constant pre-factors in the plane wave solution.

We can also consider the Lorentz gauge condition (<ref>) as an inhomogeneous
differential equation with respect to $\Lambda(x)$, i.e.

\begin{eqnarray}
\label{L-con-de}
\partial^2\Lambda(x)=\eta(x)
\end{eqnarray}

where $\eta(x)\stackrel{\text{def}}{=}-\partial_\mu A_\mu(x)$. Solving (<ref>), we obtain that

\begin{eqnarray}
\label{Sol-L-con-de}
\Lambda(x)=\Lambda_0(x)+\int d^4y \,G(x-y) \eta(y),
\end{eqnarray}

where the Green function $G(x)$ is defined as

\begin{eqnarray}
\end{eqnarray}

with the suitable regularization of operator stemmed from
the corresponding boundary conditions, see [67].

The Coulomb gauge. Using the condition $A_0^\Lambda(x)=0$ to amplify the Lorentz condition (<ref>),
we can get the Coulomb gauge condition which reads

\begin{eqnarray}
\label{C-con}
\vec{\partial}\vec{A}^\Lambda(x)= \vec{\partial}\vec{A}(x) + \Delta\Lambda(x)=0.
\end{eqnarray}

In $p$-space, the condition (<ref>) is transformed to (recall that $\vec{\partial}\vec{A}^\perp=0$
by construction)

\begin{eqnarray}
\label{C-con-p}
\vec{k}^2 a_L(k) + \vec{k}^2 \tilde\Lambda(k) = 0.
\end{eqnarray}

Again, let us study the corresponding residual gauge freedom:

\begin{eqnarray}
\label{RegC-con-1}
\vec{\partial}\vec{A}(x)=0\quad\text{and}\quad  \Delta\Lambda(x)=0.
\end{eqnarray}

For the sake of simplicity, we dwell on the case of $k^2=0$ which leads to $\vec{k}^2\neq 0$. With this,
instead of (<ref>), it is enough to stop on the equation

\begin{eqnarray}
\label{C-con-L}
\vec{k}^2 \tilde\Lambda(\vec{k})=0.
\end{eqnarray}

Hence,  the only solution of (<ref>) is $\tilde\Lambda=0$ which means that there is no any
residual freedom at all.

Therefore, in the Coulomb gauge there are no the longitudinal field components and we deal with the
physical gauge field $A^\perp_\mu$ only.

The Hamilton and axial gauges. In the similar manner, we can study the residual gauge symmetries in
the Hamilton ($A_0^\Lambda=0$) and axial ($A^{+,\,\Lambda}=0$) gauges. The residual gauge transformations
are given by the corresponding free (unfixed) gauge function $\tilde\Lambda(k)$ provided
$k_0=0$ or $k^+=0$.

§.§ Classical $SU(3)$-gauge theory (Non-abelian theory)

The next subject of our discussion is a non-Abelian gauge theory with
$SU(3)$ gauge group. In this case  case, the gauge transformation is given by

\begin{eqnarray}
\label{SU3}
A^\omega_\mu(x)=\omega(x) A_\mu(x) \omega^{-1}(x) +
\frac{i}{g}\omega(x)\partial_\mu \omega^{-1}(x)
\end{eqnarray}

which gives in the infinitesimal form

\begin{eqnarray}
\label{SU3inf}
A^{a,\,\omega}_\mu(x)= A^a_\mu(x) +
f^{abc} A^b_\mu(x) \theta^c(x) +
\frac{1}{g} \partial_\mu \theta^a(x),
\end{eqnarray}

where $\omega(x)=\exp\big( i \theta^a(x) t^a\big)$.
The decomposition of field components in the longitudinal and transverse
components is similar to the Abelian case, see above.
In contrast to the $U(1)$ gauge group, the strength tensor $G_{\mu\nu}$
is gauge-covariant. It means that all field components may change under
gauge transformations.

The Lorentz gauge. We again begin with the Lorentz gauge condition:

\begin{eqnarray}
\label{SU3inf-L-con}
&&\partial_\mu A^{a,\,\omega}_\mu(x)=
\\
&&\partial_\mu A^a_\mu(x) +
f^{abc} \partial_\mu \big( A^b_\mu(x) \theta^c(x)\big) +
\frac{1}{g} \partial^2 \theta^a(x)=0.
\nonumber
\end{eqnarray}

As above-mentioned, the gauge condition is invariant under
the residual gauge transformation:

\begin{eqnarray}
\label{ReL-con-SU-1}
\partial_\mu A^{a,\,\omega}_\mu(x)=\partial_\mu A^a_\mu(x)=0
\end{eqnarray}

or, equivalently,

\begin{eqnarray}
\label{ReL-con-SU-2}
{\cal D}_\mu^{ac}\partial_\mu \theta^c(x)=0,
\end{eqnarray}

where ${\cal D}_\mu^{ac}=\partial_\mu\delta^{ac} + g f^{abc} A^b_\mu(x)$.

In $p$-space, the condition (<ref>) takes the form of

\begin{eqnarray}
\label{ReL-con-SU-p-1}
-k^2 \theta^a(k) + i g f^{abc} k_\mu A^{b,\,L}_\mu(k) \theta^c(k)=0.
\end{eqnarray}

If $k^2=0$ and, therefore, $k_\mu A^{b,\,L}_\mu(k)=0$, then the gauge function $\theta(x)$ cannot be fixed
and generates the residual gauge transformation.

The Hamilton gauge. The similar situation occurs in the Hamilton gauge,
$A^\omega_0=0$. The residual transformation is induced by the gauge function which obeys

\begin{eqnarray}
\label{ReH-con}
\partial_0 \theta^a(x_0,\vec{x})=0.
\end{eqnarray}

Hence, the solution of this equation is rather trivial: $\theta$-function is the time-independent function, $\theta_0(\vec{x})$.

In the momentum representation, the condition (<ref>) gives us the equation

\begin{eqnarray}
\label{ReH-con-p-1}
\int (d^4k)e^{ikx} k_0 \theta^a(k_0,\vec{k})=0
\end{eqnarray}

which has a solution as

\begin{eqnarray}
\label{Sol-ReH-con-p}
\theta^a_0(k)=\delta(k_0)\theta^a_0(\vec{k}).
\end{eqnarray}

Therefore, we find in the coordinate representation

\begin{eqnarray}
\int (d^4k)e^{ikx} \delta(k_0)\theta^a_0(\vec{k}) = \theta^a_0(\vec{x})
\end{eqnarray}

which coincides with the results of the preceding paragraph.

The axial gauge. Working in the axial gauge, $A^{+,\,\omega}=0$, in the similar manner
we are able to find the gauge function that is responsible for the residual gauge symmetry.
We impose the condition

\begin{eqnarray}
\label{RecAx-con-1}
\end{eqnarray}

or, in the equivalent form,

\begin{eqnarray}
\label{RecAx-con-2}
\partial^{+}\theta^a(x^+,x^-,\vec{\bf x}_\perp)=0\quad \text{with}\quad
\partial^+=\partial_-=\frac{\partial}{\partial x^-}.
\end{eqnarray}

The solution of this trivial differential equation is the $x^-$-independent function
$\theta^a_0(x^+,\vec{\bf x}_\perp)$ which has the following form in $p$-space
(cf. (<ref>) and (<ref>)):

\begin{eqnarray}
\label{ReAx-con-p-1}
\theta^a_0(k^+,k^-,\vec{\bf k}_\perp)=\delta(k^+)\theta^a_0(k^-,\vec{\bf k}_\perp)
\end{eqnarray}

where $\theta^a_0(k^-,\vec{\bf k}_\perp)$ is an arbitrary gauge function related to the residual symmetry.

It is instructive to focus on the finite gauge transformations and corresponding gauge condition, namely

\begin{eqnarray}
\label{ReAx-f-con}
\\
&&\omega(x) A^+(x) \omega^{-1}(x) +
\frac{i}{g}\omega(x)\partial^+ \omega^{-1}(x)=0.
\nonumber
\end{eqnarray}

The solution of this equation can easy be found, it reads

\begin{eqnarray}
\label{Sol-Ax-con-f}
\omega_0(x)=\mathbb{P}{\rm exp}\Big\{ ig \int\limits^{x^-}_{\mathds{C}}
dz^- A^+(x^+,z^-,\vec{\bf x}_\perp) \Big\}
\end{eqnarray}

where, generally speaking, $\mathds{C}$ is an arbitrary constant.
We stress that the solution $\omega_0(x)$ is valid for $\forall x \in \mathds{R}^4$.
At the same time, this function can be multiplied by an arbitrary $x^-$-independent
gauge function to produce another solution of equation (<ref>), i.e.

\begin{eqnarray}
\label{Omega-gen}
W(x^+,x^-,\vec{\bf x}_\perp) = \bar\omega(x^+,\vec{\bf x}_\perp) \omega_0(x^+,x^-,\vec{\bf x}_\perp),
\end{eqnarray}

where $\bar\omega(x^+,\vec{\bf x}_\perp)=\exp\big( i\theta^a(x^+,\vec{\bf x}_\perp)t^a \big)$.
Indeed, one can demonstrate that $A^{+,\,W}(x)=0$.

To study the residual symmetry, we have to demand that $A^+(x)=0$ for any $x$.
Therefore, from (<ref>), we obtain that the function

\begin{eqnarray}
\label{Omega-gen-res}
W(x^+,x^-,\vec{\bf x}_\perp)\Big|_{A^+=0} = \bar\omega(x^+,\vec{\bf x}_\perp)
\end{eqnarray}

generates the residual transformation we are interested in.

Let us now return to the gauge function presented by (<ref>).
The case of $k^+=0$ (which provides us the residual symmetry) leads to the
so-called spurious singularity in the gluon propagator in the axial gauge, see the next subsection.
If we adopt a procedure to regularize this singularity with the help of some well-defined procedure,
$[ k^+]_{reg}\neq 0$, then the existence condition for the residual symmetry, see (<ref>),
has to be given by (in the momentum representation)

\begin{eqnarray}
\label{ReAx-con-reg}
\int (d^4k)e^{ikx} [k^+]_{reg} \delta(k^+)\theta^a_0(k^-,\vec{\bf k}_\perp)=0.
\end{eqnarray}

Hence, the only possibility to satisfy this equation is to demand that $\theta^a_0(k^-,\vec{\bf k}_\perp)=0$ which
means that we fix the remaining residual symmetry.
Thus, we conclude that the spurious singularity is fixed if and only if we do not have the residual gauge symmetry.
On the other hand, we may say that the residual gauge fixing is enough for the elimination of spurious singularity.

§.§ Spurious singularity of gluon propagator

Let us return to the issue of the spurious singularity which appears in the gluon propagator in
the axial gauge $A^+=0$.

The generating functional for gluons (gluonodynamics)
in the most general gauge $F(A^\theta)=0$

\begin{eqnarray}
\label{Z}
&&\mathbb{Z}=N \int {\cal D} A_\mu e^{iS[A]} =
\\
&&\tilde N \int {\cal D} A_\mu \, \Delta_c[A]\, \delta\big(F(A)\big) \, e^{iS[A]},
\nonumber
\end{eqnarray}

where $\tilde N$ involves the infinite gauge group volume, $\int d\theta$, and we use

\begin{eqnarray}
\mathds{1}= \int d\theta \Delta_c[A] \delta\big( F(A^\theta)\big),
\quad \Delta_c[A^\theta] =  \Delta_c[A].
\end{eqnarray}

Instead of solving the gauge condition $F(A^\theta)=0$ with respect to the group function
$\theta$ within the generalized Hamilton formalism, we separate out the infinite group volume, $\int d\theta$,
in the generating functional (the Faddeev-Popov approach).

The next trick is related to the exponentiation of $\delta\big(F(A)\big)$. We introduce the generalized gauge condition as
$F(A)=C$ with $\delta C/\delta A_\mu=0$.  The generalizing functional $\mathbb{Z}$ must be independent on $C$.
Therefore, to get the $C$-independent functional we have to integrate out over this parameter $C$.
Using the integration measure defined as

\begin{eqnarray}
dC \exp\big( - \frac{i}{2\xi} \int d^4 x \,  C^2(x) \big),
\end{eqnarray}

we have

\begin{eqnarray}
\label{Z-2}
&&\hspace{-0.5cm}\mathbb{Z}=\tilde N \int dC e^{\big( - \frac{i}{2\xi} \int d^4 x C^2(x) \big)}
\int {\cal D} A_\mu \, \Delta_c[A]\, \delta\big(F(A)-C\big) \, e^{iS[A]}
\nonumber\\
&&=\tilde N
\int {\cal D} A_\mu \, \Delta_c[A] \, e^{iS[A]- \frac{i}{2\xi} \int d^4 x F^2(A)}.
\end{eqnarray}

In (<ref>), the effective action  with the gauge-fixing term,

\begin{eqnarray}
S_{\text{fix}}=- \frac{1}{2\xi} \int d^4 x F^2(A),
\end{eqnarray}

is now not gauge-invariant anymore. As a result of this trick, we don't need to solve
the gauge condition with respect  to the gauge function.

Let the gauge condition $F(A)=0$ be $A^+=0$ with $n^2=0$. In this case, the determinant $\Delta_c[A]$ is independent on $A$
and, therefore, we are able to include this determinant in the normalization of functional.
Thus, the effective Lagrangian reads

\begin{eqnarray}
\label{Leff}
{\cal L}_{\text{eff}}= -\frac{1}{4} G_{\mu\nu}G_{\mu\nu} - \frac{1}{2\xi} \big( n\cdot A \big)^2.
\end{eqnarray}

This Lagrangian yields the effective action which can be written as

\begin{eqnarray}
\label{Seff}
S_{\text{eff}}= \frac{1}{2} \int d^4x\, A_\mu(x)\, K_{\mu\nu}(x)\, A_\nu(x),
\end{eqnarray}

\begin{eqnarray}
\label{Kop}
K_{\mu\nu}(x)=g_{\mu\nu}\partial^2 - \partial_\mu \partial_\nu - \frac{1}{\xi} n_\mu n_\nu.
\end{eqnarray}

In $p$-space, the operator $K_{\mu\nu}$ has an inverse operator which, in the limit of $\xi\to 0$, is given by

\begin{eqnarray}
\label{InverseK}
\nonumber\\
&&d_{\mu\nu}(k,n)=g_{\mu\nu}-\frac{k_\mu n_\nu + k_\nu n_\mu}{k^+}.
\end{eqnarray}

As we have demonstrated in the preceding subsection, when we fix/regularize
the spurious singularity $[k^+]_{reg}$ it means that we fix the residual gauge
symmetry defined by the gauge function $\theta^a(k^-,\vec{\bf k}_\perp)$ and vise versa.

We also remind that it is not possible to fix the residual gauge simply by means of adding of

\begin{eqnarray}
\label{Amin-Leff}
\frac{1}{2\xi_2}\big( n^*\cdot A \big)^2
\end{eqnarray}

in Eqn. (<ref>). In this case, the inverse kinematical operator (see, Eqn. (<ref>)) does not
exist due to the fact that the free (without the coefficients) tensors $n_{\mu}n_{\nu}$ and
$n^*_{\mu}n^*_{\nu}$ present in the corresponding equation to determine the coefficients.
Indeed, introducing the Lorentz parametrization (where the coefficients have to be determined)

\begin{eqnarray}
\label{Lor-par-d}
&&d_{\nu\rho}(k,n,n^*)=g_{\nu\rho} + a_1 k_{\nu}k_{\rho}+
b_2 k_{\nu}n_{\rho}+
b_3 n_{\nu}k_{\rho}+
\nonumber\\
&&b_4 k_{\nu}n^*_{\rho}+
b_5 n^*_{\nu}k_{\rho}+
c_6 n_{\nu}n_{\rho}+
c_7 n^*_{\nu}n^*_{\rho},
\end{eqnarray}

\begin{eqnarray}
\text{dim}_M[a_1]=-2,\, \text{dim}_M[b_i]=-1,\, \text{dim}_M[c_j]=0,
\end{eqnarray}

the contraction equation on the coefficients (or, in other words, the Green function equation)

\begin{eqnarray}
\label{Cont-Eqn}
\end{eqnarray}

involves the tensors $n_{\mu}n_{\nu}$ and $n^*_{\mu}n^*_{\nu}$ which stay without
coefficients. It means that the inverse operator cannot be derived.

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