Digitally Restricted Sets and the Goldbach Conjecture: An Exceptional Set Result

TL;DR

The number of even integers in $\mathcal {A}$ which are less than X and not representable as the sum of two primes is less than $|\mathcal {A}\cap \{1,\ldots,X\}|^{1-\delta }$ .

Abstract

We show that for any set $D$ of at least two digits in a given base $b$, there exists a $δ(D,b)>0$ such that within the set $\mathcal{A}$ of numbers whose digits base $b$ are exclusively from $D$, the number of even integers in $\mathcal{A}$ which are less than $X$ and not representable as the sum of two primes is less than $|\mathcal{A}(X)|^{1-δ}$

Theorems & Definitions (9)

• Theorem 1.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 4.1
• proof
• Lemma 4.2
• proof