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Alf van der Poorten

CeNTRe for Number Theory Research (Australia)

An awful problem about integers in base four

Back in the early eighties, John Loxton and I had our attention captured by a question of Brown and Moran concerning a construction seemingly relevant to work that eventually led to their 1984 paper with Tijdeman: “Riesz products are basic measures”. In brief, one may of course write the integers in base four using 0, 1, and 2 as the four digits. Can every integer be expressed as a quotient of integers requiring just the three digits 0, and 1? Such questions are notoriously slippery and John and I therefore remain inordinately proud of the clever tricks we eventually recalled, after far too much effort, to settle the issue. I will tell the story of our travails.