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Let { s j } j = 1 n \left \{ s_{j}\right \} _{j=1}^{n} be positive integers. We show that for any 1 ≤ L ≤ n , 1\leq L\leq n, ‖ ∏ j = 1 n ( 1 − z s j ) ‖ L ∞ ( | z | = 1 ) ≥ exp ( 1 2 e L ( s 1 s 2 … s L ) 1 / L ) . \begin{equation*} \left \Vert \prod _{j=1}^{n}\left ( 1-z^{s_{j}}\right ) \right \Vert _{L_{\infty }\left ( \left \vert z\right \vert =1\right ) }\geq \exp \left ( \frac {1}{2e}\frac {L}{\left ( s_{1}s_{2}\ldots s_{L}\right ) ^{1/L}}\right ) . \end{equation*} In particular, this gives geometric growth if a positive proportion of the { s j } \left \{ s_{j}\right \} are bounded. We also show that when the { s j } \left \{ s_{j}\right \} grow regularly and faster than j ( log j ) 2 + ε j\left ( \log j\right ) ^{2+\varepsilon } , some ε > 0 \varepsilon >0 , then the norms grow faster than exp ( ( log n ) 1 + δ ) \exp \left ( \left ( \log n\right ) ^{1+\delta }\right ) for some δ > 0 \delta >0 .
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Singmaster’s Conjecture In The Interior Of Pascal’s Triangle
Abstract Singmaster’s conjecture asserts that every natural number greater than one occurs at most a bounded number of times in Pascal’s triangle; that is, for any natural number $$t \geq 2$$, the number of solutions to the equation $$\binom{n}{m} = t$$ for natural numbers $$1 \leq m \lt n$$ is bounded. In this paper we establish this result in the interior region $$\exp(\log^{2/3+\varepsilon} n) \leq m \leq n - \exp(\log^{2/3+\varepsilon} n)$$ for any fixed ɛ > 0. Indeed, when t is sufficiently large depending on ɛ, we show that there are at most four solutions (or at most two in either half of Pascal’s triangle) in this region. We also establish analogous results for the equation $$(n)_m = t$$, where $$(n)_m := n(n-1) \dots (n-m+1)$$ denotes the falling factorial.
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- PAR ID:
- 10371470
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- The Quarterly Journal of Mathematics
- Volume:
- 73
- Issue:
- 3
- ISSN:
- 0033-5606
- Format(s):
- Medium: X Size: p. 1137-1177
- Size(s):
- p. 1137-1177
- Sponsoring Org:
- National Science Foundation
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