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Daniel Larsen

Publications and source records attributed to Daniel Larsen.

5 recordsLinked to original sources
arXiv2026

Three Questions of Erd\H{o}s-Nathanson on Asymptotic Bases of Order 2

We study three natural properties that measure the robustness of asymptotic bases of order 2: having divergent representation function, being decomposable as a union of two bases, and containing a minimal basis. Erd\H{o}s and Nathanson showed that sufficiently rapid growth of the representation function (specifically, $r_A(n) \ge C \log n$ for appropriate $C$) implies both decomposability and the existence of a minimal basis. We prove that for weaker growth rates, these three properties are independent. The construction proceeds via an inductive scheme on exponentially growing intervals.

math.NT↗
arXiv2026

Robust additive bases without minimal subbases

There exists a set $A$ of positive integers such that the number of representations of a large positive integer $m$ as a sum of two elements of $A$ grows with a lower bound of order $\log m$, but for which there is no subset $D$ of $A$ minimal for the property that $D+D$ contains all sufficiently large positive integers.

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arXiv2025

Carmichael Numbers in All Possible Arithmetic Progressions

We prove that every arithmetic progression either contains infinitely many Carmichael numbers or none at all. Furthermore, there is a simple criterion for determining which category a given arithmetic progression falls into. In particular, if $m$ is any integer such that $(m,2\phi(m))=1$ then there exist infinitely many Carmichael numbers divisible by $m$. As a consequence, we are able to prove that $\liminf_{n\text{ Carmichael}}\frac{\phi(n)}{n}=0$, resolving a question of Alford, Granville, and Pomerance.

math.NT↗
arXiv2021

Bertrand's Postulate for Carmichael Numbers

Alford, Granville, and Pomerance proved that there are infinitely many Carmichael numbers. In the same paper, they ask if a statement analogous to Bertrand's postulate could be proven for Carmichael numbers. In this paper, we answer this question, proving the stronger statement that for all $\delta>0$ and $x$ sufficiently large in terms of $\delta$, there exist at least $e^{\frac{\log x}{(\log\log x)^{2+\delta}}}$ Carmichael numbers between $x$ and $x+\frac{x}{(\log x)^{\frac{1}{2+\delta}}}$.

math.NT↗