Arithmetic properties of cubic and biquadratic theta series

TL;DR

This paper investigates the algebraic nature of cubic and biquadratic theta series, showing they are not algebraic numbers when evaluated at reciprocals of integers, using advanced linear independence techniques and recent Waring's problem results.

Contribution

It improves previous results by proving these theta series are not algebraic numbers at reciprocals of integers, employing a novel nested gaps technique and recent Waring's problem findings.

Findings

Cubic and biquadratic theta series are not algebraic numbers at reciprocals of integers.

Introduction of a nested gaps technique for linear independence.

Application of recent Waring's problem results to theta series analysis.

Abstract

A cubic (resp. biquadratic) theta series is a power series whose n-th coefficient is equal to 1 if n is a perfect cube (resp. fourth power) and zero otherwise. We improve on a result of Bradshaw by showing that such series is not a cubic (resp. biquadratic) algebraic number when evaluated at reciprocals of integers. The proof relies on a "nested gaps technique" for linear independence and on recent results by the author on Waring's problem for cubes and biquadrates.