2 Preliminaries

We first introduce notation. As is common in analytic number theory, let $e(\alpha ):=e^{2\pi i\alpha }$ and let the symbol $f(x)\ll g(x)$ mean that there exists a positive constant c such that $|f(x)|\leq cg(x)$ after restricting to an appropriate domain. Whenever we use the letter $\varepsilon $ , we mean that the statement is true for every $\varepsilon>0$ . Throughout this paper, the letter p always denotes primes. Take k, the number of digits, to be sufficiently large, and let $X=b^k$ , so that $\mathcal {A}_k$ has only values less than X.

When discussing digitally restricted sets we need to deal with the technicality of leading zeros. We want to study all digitally restricted numbers less than X, of any number of digits, but when $0\not \in D$ we see that $\mathcal {A}_k$ only contains numbers with precisely k digits and when $j< k$ we have $\mathcal {A}_j\cap \mathcal {A}_{k}=\emptyset $ , in contrast to the case where $0\in D$ , in which a j-digit number is also a k-digit number with $k-j$ leading zeros and $\mathcal {A}_j\subset \mathcal {A}_k$ . When $0\in D$ we may show that $|\mathcal {A}_k\cap E(X)|$ is small to obtain the desired result, but when $0\not \in D$ we note that $|\mathcal {A}\cap E(X)| = \sum _{j\leq k}|\mathcal {A}_k\cap E(X)|$ , so proving the result for $\mathcal {A}_k$ will prove it for $\mathcal {A}(X)$ in either case, and we will henceforth deal only with $\mathcal {A}_k$ .

Now we define the generating function for $\mathcal {A}_k$ to be

$$ \begin{align*}f(\alpha) := \sum_{n\in\mathcal{A}_k} e(n\alpha).\end{align*} $$

The generating function for primes on the other hand will require a parameter. Let $\delta _0$ be a sufficiently small constant, which may depend on b, to be chosen later. We define

$$ \begin{align*}S(\alpha) := \sum_{X^{6\delta_0}<p<X} e(p\alpha)\log p.\end{align*} $$

For any ${\mathfrak B}\subset [0,1]$ and any even $n<X$ , we define

$$ \begin{align*}r(n,{\mathfrak B}):=\int_{\mathfrak B} S(\alpha)^2e(-n\alpha)\,d\alpha.\end{align*} $$

By orthogonality, $r(n,[0,1])$ counts solutions to $p_1+p_2=n$ with weight $\log p_1\log p_2$ . To evaluate this integral we apply the Hardy–Littlewood circle method. Define the major arcs to be

(2.1) $$ \begin{align} \mathfrak{M}_{\delta_0} := \bigcup_{0<q<X} \{\alpha\in [0,1]:\|q\alpha\|<X^{6\delta_0-1}\} \end{align} $$

and the minor arcs to be

(2.2) $$ \begin{align}\mathfrak{m}_{\delta_0} := [0,1]\setminus \mathfrak{M}_{\delta_0}. \end{align} $$

In Section 3 we establish Lemmas 3.1 and 3.2, the facts about $f(\alpha )$ and $\mathcal {A}_k$ which we will use in the proof. In Section 4 we combine Section 3 with commonly known facts about S to prove Theorem 1.1.

3 Properties of digital restrictions

In this section we prove the properties of digitally restricted sets which we will need to show Theorem 1.1. The two results we need are Lemma 3.1, a mean value estimate, and Lemma 3.2, a bound on how many elements of $\mathcal {A}_k$ can be multiples of a given large value.

Lemma 3.1. For all $\delta>0$ , there exists a $t_0=t_0(b,\delta )$ such that whenever $t\geq t_0$ ,

$$ \begin{align*}\int_0^1 |f(\alpha)|^t\,d \alpha \ll |\mathcal{A}_k|^tX^{\delta-1}.\end{align*} $$

Proof. We prove the bound by exploiting the independence of digits of values in $\mathcal {A}_k$ . By treating each digit as its own variable, the number of variables goes to infinity while the number of values any variable can take remains constant.

Note that by orthogonality, if $t=2s$ is an even integer, the integral counts solutions to the equation

$$ \begin{align*} \sum_{i=1}^{s}(x_i-y_i)=0 \end{align*} $$

with $x_i,y_i\in \mathcal {A}_k$ . We now interpret each $x_i$ and $y_i$ as a linear combination of k variables, each corresponding to a digital place. Let

$$ \begin{align*}x_i = \sum_{j=0}^{k-1} x_{i,j}b^j\end{align*} $$

and the same for $y_i$ . We are now counting solutions to

(3.1) $$ \begin{align} \sum_{j=0}^k b^j\sum_{i=1}^{s}(x_{i,j}-y_{i,j})=0 \end{align} $$

with $x_{i,j},y_{i,j}\in D$ .

First, if every allowed digit shares a common factor then we know that every $x_{i,j}$ and $y_{i,j}$ will share that common factor, permitting us to divide everything by that common factor. Thus, we may assume that the allowed digits do not share a common factor. To count solutions to (3.1) we investigate what addition looks like in base b, and in particular we look at each digital place. Let

$$ \begin{align*}\sigma_{j} : = \sum_{i=1}^s (x_{i,j}-y_{i,j})\end{align*} $$

and

$$ \begin{align*}\tau_{j} := \sum_{l<j}b^l\sigma_{l}.\end{align*} $$

Now, we note that for (3.1) to hold, we require that it holds modulo b, and thus $\sigma _0\equiv 0\,\mod b$ , or to be precise, the 1s digits must add up to 0 mod b. Further, we know that (3.1) holds mod $b^j$ for any j, meaning that $\tau _{j} \equiv -b^j \sigma _{j}\,\mod b^{j+1}$ . After we have chosen the digits up to j we will thus require that $\sigma _{j} \equiv n$ mod b for some fixed n which is determined by the digits up to j.

No matter what the digits in the other places are, we will thus have at most

$$ \begin{align*}\max_{n<b}\#\{(x_{0,j},x_{1,j},\ldots,x_{s,j},y_{0,j},\ldots,y_{s,j}):\sigma_{j}\equiv n\,\mod b\}\end{align*} $$

choices for how to assign the jth digits. By orthogonality, we can count these using the function

$$ \begin{align*} u(n,b,D):=\frac{1}{b}\sum_{0\leq a<b} \bigg(\bigg|\sum_{d\in D}e\bigg(\frac{da}{b}\bigg)\bigg|^{2s}e\bigg(-\frac{na}{b}\bigg)\bigg). \end{align*} $$

Noting that this does not depend on j, we thus have an upper bound on the number of solutions to (3.1) which is the product of the upper bounds on the sums of each digit, which is bounded by the maximum of $u(n,b,D)^k$ over all possible n. Finally, noting that by the lack of a common factor of acceptable digits, for all nonzero a,

$$ \begin{align*}\bigg|{\sum_{d\in D}e\bigg(\frac{da}{b}\bigg)}\bigg|<|D|.\end{align*} $$

Dividing both sides by $|D|$ and noting that there are finitely many nonzero a then yields

(3.2) $$ \begin{align}\max_{0<a<b}\frac{1}{D} \bigg|{\sum_{d\in D}e\bigg(\frac{da}{b}\bigg)}\bigg| <1.\end{align} $$

We now choose s to be at least

$$ \begin{align*}\frac{\log(\delta/a)}{\log\big({\max_{0<a<b}\tfrac{1}{D}|{\sum_{d\in D}e\big(\tfrac{da}{b}\big)}|}\big)}.\end{align*} $$

By (3.2), this must be a finite constant depending only on b, D and $\delta $ , which makes it a valid choice of s. Summing over all values of a then yields

$$ \begin{align*}\sum_{0\leq a<b} \bigg({\frac{1}{D}\bigg|{\sum_{d\in D}e\bigg(\frac{da}{b}\bigg)}\bigg|}^{2s}\bigg)<1+\delta.\end{align*} $$

Rearranging this and noting that since $|D|\geq 2$ , we have $|D|^{\delta }>1+\delta $ , we can see that

$$ \begin{align*}u(b;D)=\frac{1}{b}\sum_{0\leq a<b}\bigg|{\sum_{d\in D}e\bigg(\frac{da}{b}\bigg)}\bigg|^{2s} \leq \frac{(1+\delta_s)}{b}|D|^{2s}\leq \frac{1}{b}|D|^{2s+\delta}.\end{align*} $$

Taking the product across all digital places now yields the desired result.

Lemma 3.2. For any m, we have

$$ \begin{align*}\#\{n\in\mathcal{A}_k:m \mid n\}\leq \frac{b|\mathcal{A}_k|}{b^{-1}+(m/2)^{\log|D|/\!\log b}} \ll \frac{|\mathcal{A}_k|}{m^{\log|D|/\!\log b}}.\end{align*} $$

Proof. Note that for every integer $n\in \mathcal {A}_k$ which is a multiple of m, we may change the last $\lfloor {\log (m/2)}/{\log b}\rfloor $ many digits and obtain $|D|^{\lfloor {\log (m/2)}/{\log b}\rfloor } \geq b^{-1}(m/2)^{{\log |D|}/{\log b}}$ many other values in $\mathcal {A}_k$ which are less than $m/2$ away from n. Since any integer may be less than $m/2$ from at most one multiple of m, this provides a map from values in $\mathcal {A}_k$ which are multiples of m to disjoint sets of at least $b^{-1}(m/2)^{{\log |D|}/{\log b}}$ many integers in $\mathcal {A}_k$ which are not multiples of m. Taking the disjoint union of all these sets, along with the associated multiples of m, returns a subset of $\mathcal {A}_k$ , and hence the sum of the cardinalities must be at most $|\mathcal {A}_k|$ . More formally, we see that

$$ \begin{align*}\sum\limits_{\substack{n\in\mathcal{A}_k\\ m|n}}1+b^{-1}(m/2)^{{\log|D|}/{\log b}} \leq |\mathcal{A}_k|.\end{align*} $$

and hence, dividing by the term being summed, we see that

$$ \begin{align*}\sum\limits_{\substack{n\in\mathcal{A}_k\\ m|n}} 1 \leq \frac{b|\mathcal{A}_k|}{b^{-1}+(m/2)^{\log|D|/\!\log b}},\end{align*} $$

which is the desired result.

Since the expected number of multiples of m lying in $\mathcal {A}_k$ would be $|\mathcal {A}_k|/m$ , it is natural to ask whether we can obtain this or possibly $|\mathcal {A}_k|/m^c$ for some absolute c which does not depend on b and $|D|$ . Unfortunately, this is ruled out by those m which are multiples of powers of b. Taking $m=b^j$ , we see that $n\in \mathcal {A}_k$ being a multiple of m is equivalent to the last j digits being 0, and that when 0 is an allowed digit there will be $N/N^{j/k}=N/m^{\log |D|/\!\log b}$ of these. For values m which are coprime to b, a significant amount of research has been done (see, for example, [Reference Konyagin5]), but this only applies to small values of m and not the large values we will need, about which few positive results are known. Konyagin [Reference Konyagin5] showed that we cannot hope for too much, as there exist relatively small coprime moduli, of the order of $\exp (c\log X/\!\log \log X)$ , in which $\mathcal {A}$ is not evenly distributed between residue classes. Despite not being evenly distributed, it is still expected that $\mathcal {A}$ is not too badly distributed. In particular, Konyagin conjectured that $\mathcal {A}(X)$ meets every residue class modulo m whenever $(m,b)=1$ and $m<X^c$ for some constant $c=c(b)$ .

4 Exceptional sets in Goldbach’s conjecture

In this section, following [Reference Brüdern, Kawada and Wooley1], we use what we have proven about numbers with restricted digits to prove Theorem 1.1. We now study $r(n,{\mathfrak B})$ as we defined it in Section 2. In particular, we study $r(n,\mathfrak {M})$ and $r(n,\mathfrak {m})$ as defined in (2.1) and (2.2), respectively. Our goal will be to show, for almost all n in $\mathcal {A}_k$ , that $r(n,\mathfrak {m})\ll X^{1-\delta _1}$ and $r(n,\mathfrak {M})\gg X^{1-\delta _2}$ with $\delta _1<\delta _2$ . First, we handle the minor arcs.

Lemma 4.1. There exists $\delta _1=\delta _1(b,D)>0$ such that for all sufficiently large X, there exists a set $B\subset \mathcal {A}_k$ with $|B|\ll |\mathcal {A}_k|X^{-\delta _1}$ , and for all even n in $\mathcal {A}_k\setminus B$ we have

$$ \begin{align*}|r(n,\mathfrak{m})|\ll |\mathcal{A}_k|X^{-\delta_1}\end{align*} $$

Proof. By symmetry of $\mathfrak {m}$ , we see that $r(n,\mathfrak {m})$ is real. Set

$$ \begin{align*}\eta(n):=\mathrm{sign}(r(n,\mathfrak{m})).\end{align*} $$

Now, we let

$$ \begin{align*}K(\alpha):= \sum\limits_{\substack{0<n<X\\n\in\mathcal{A}}} \eta(n)e(-n\alpha)\end{align*} $$

and rewrite the expression we are trying to bound as

$$ \begin{align*}\sum_{n\in \mathcal{A}_k} |r(n,\mathfrak{m})| = \int_{\mathfrak{m}} S(\alpha)^2 K(-\alpha)\,d \alpha.\end{align*} $$

From Hölder’s inequality, this is at most

$$ \begin{align*} \bigg(\sup_{\alpha\in\mathfrak{m}}|S(\alpha)|\bigg)^{2/t}\bigg(\int_0^1|S(\alpha)|^2\,d \alpha\bigg)^{1-1/t}\bigg(\int_0^1|K(-\alpha)|^t\,d \alpha\bigg)^{1/t}, \end{align*} $$

and we can now estimate each term individually.

First, by [Reference Vaughan12],

$$ \begin{align*}\sup_{\alpha\in\mathfrak{m}}|S(\alpha)| \ll X^{1-3\delta_0}(\log X)^4.\end{align*} $$

By applying orthogonality, we trivially have

$$ \begin{align*}\int_0^1 |S(\alpha)|^2\,d \alpha \ll \sum_{P<p<X}(\log p)^2 \ll X\log X.\end{align*} $$

Now we estimate the contribution of K. Observe that for integer values of t, the mean value $\int _0^1 |K(\alpha )|^{2t}\,d \alpha $ counts solutions with $x_i,y_i\in \mathcal {A}_k$ to the additive equation

$$ \begin{align*}\sum\limits_{\substack{i\leq t}}(x_i-y_i) = 0,\end{align*} $$

with each solution being weighted with $\pm 1$ . Thus, it is clearly less than or equal to the unweighted version, namely $\int _0^1 |f(\alpha )|^{2t}\,d \alpha $ , which when we take t sufficiently large is itself $O(|\mathcal {A}_k|^{2t}X^{\delta _0-1})$ by Lemma 3.1.

Thus, in total,

$$ \begin{align*} \int_0^1S(\alpha)^2K(-\alpha)\,d \alpha &\ll |\mathcal{A}_k|X^{1-1/t+(2-6\delta_0)/t+(-1+\delta)/t}(\log X)^{8/t+1-1/t}\\ &=|\mathcal{A}_k|X^{1-5\delta_0/t}(\log X)^{1+7/t} \end{align*} $$

implying that

$$ \begin{align*}\sum\limits_{\substack{n<X\\n\in\mathcal{A}}}|r(n,\mathfrak{m})|\ll NX^{1-2\delta_2}\end{align*} $$

for some $\delta _2>0$ . Now using the trivial bound, that

$$ \begin{align*}|\{n\in\mathcal{A}_k:|r(n,\mathfrak{m})|>X^{1-\delta_1}\}|\leq X^{\delta_1-1}\sum_{n\in |\mathcal{A}_k|}|r(n,\mathfrak{m})|,\end{align*} $$

we obtain the desired result. We note that we just bounded the number of values in $\mathcal {A}$ with a large minor arc, and the number of even values in $\mathcal {A}$ with a large minor arc is clearly even less.

Next we deal with the major arcs.

Lemma 4.2. Given Y such that $1\leq Y\leq |\mathcal {A}_k|^{\delta _0}$ , we have

$$ \begin{align*}r(n,\mathfrak{M}) \gg XY^{-1/2}(\log X)^{-1}\end{align*} $$

for all even $n\in \mathcal {A}_k(X)$ except for at most $O(|\mathcal {A}_k|^{1+\varepsilon }Y^{-(\log |D|)/\!\log b})$ many n.

Proof. This follows immediately from the proof of Lemma 2 in [Reference Brüdern, Kawada and Wooley1]. Note that the only step in the proof which depends on the set being counted occurs in the display following (18) of the latter paper, where numbers n such that $(n,\tilde {r})>Y$ were discarded. We replace that with

$$ \begin{align*}\sum\limits_{\substack{d|\tilde{r}\\ d>Y}}\sum\limits_{\substack{n\in \mathcal{A}_k\\ D|n}} 1\ll \sum\limits_{\substack{d|\tilde{r}\\ d>Y}} (|\mathcal{A}_k|/d^{\log|D|/\!\log b}) \ll |\mathcal{A}_k|^{1+\varepsilon}Y^{-\log|D|/\!\log b}.\end{align*} $$

If we discard $O(|\mathcal {A}_k|^{1+\varepsilon }Y^{-\log |D|/\!\log b})$ many numbers, the rest of the proof follows.

Now that we have both the major arcs and the minor arcs, we are equipped to prove Theorem 1.1. Taking Y to be less than $X^{\delta _1/3}$ , we see that there is some set B such that for all $n\in B$ , we have $r(n,\mathfrak {m})=o(r(n,\mathfrak {M}))$ and

$$ \begin{align*}|\mathcal{A}\setminus B| \ll|\mathcal{A}_k|^{1+\varepsilon}X^{-(\delta_1\log|D|)/(3\log b)}+|\mathcal{A}_k|X^{-\delta_1}.\end{align*} $$

Recalling that the number of solutions to (1.1) is

$$ \begin{align*}r(n,[0,1])=r(n,\mathfrak{m})+r(n,\mathfrak{M}) \gg r(n,\mathfrak{M})> 0,\end{align*} $$

for all n other than those exceptions, shows that the Goldbach conjecture holds for all other values in $\mathcal {A}_k$ .