Diophantine Approximation with Products of Two Primes

TL;DR

This paper proves that for any irrational number and certain approximation rates, infinitely many semiprimes can approximate the number closely, and also constructs large palindromes with exactly two prime factors.

Contribution

It establishes new bounds on Diophantine approximation using products of two primes and constructs large palindromes with two prime factors in various bases.

Findings

Infinitely many semiprimes approximate irrationals within n^{-8/23}.

Existence of large 3-digit palindromes with exactly two prime factors.

New bounds on approximation rates for products of two primes.

Abstract

We show that for any irrational $\alpha$ and any $\tau<8/23$ there are infinitely many $n$ which are the product of two primes for which $$\|n\alpha\|\leq n^{-\tau}.$$ We also show that for all sufficiently large $b$ there exist 3-digit palindromes in base $b$ with precisely two prime factors.