Line counts and a census for the exceptions to $\lVert(3/2)^n\rVert\ge(3/4)^n$, with an application to Waring's problem

Abstract

Problem 10.13 of Bugeaud's \emph{Distribution modulo one and Diophantine approximation} asks for an upper bound for the number of positive integers $n$ with $\lVert(3/2)^n\rVert<(3/4)^n$, where $\lVert\cdot\rVert$ denotes the distance to the nearest integer. Mahler proved in 1957 that there are only finitely many such $n$; his proof bounds neither their size nor their number. We prove that the exceptions with $n\ge10$ lie on at most $K(\varepsilon^{*})=1\,856\,360\,182\,227<1.86\cdot10^{12}$ lines of $\mathbb{Q}^{2}$, by feeding the associated $S$-unit points into the quantitative Ridout theorem of Bugeaud and Evertse at the sharp exponent $\varepsilon^{*}=\log(4/3)/\log3$, whose exponent budget the problem meets with equality. Counting the solutions on a single line is an open problem; conditional on a bound $H$ for the number of exceptions per line, the total number of exceptions is at most $5+H\cdot\Kb(\varepsilon^{*})$, and the ideal Waring formula $g(n)=2^{n}+\lfloor(3/2)^{n}\rfloor-2$ holds for all $n\ge2$ with at most $H\cdot1.88\cdot10^{12}$ exceptions, with no additive constant. A kernel-certified census determines the exceptions up to $256$ as $\{1,2,3,4,7\}$ and their line structure as the partition $\{1\},\{2,3\},\{4\},\{7\}$, and a stratified variant shows that bases of proportionally tall towers lie on fewer lines, at most $5.38\cdot10^{11}$ for spans of a quarter of the base. All results are formally verified in Lean~4.

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