Let $k$ be a perfect field of positive characteristic and let $X$ be a smooth irreducible quasi-compact scheme over $k$. The Drinfeld–Kedlaya theorem states that for an irreducible $F$-isocrystal on $X$, the gap between consecutive generic slopes is bounded by one. In this note we provide a new proof of this theorem. Our proof utilizes the theory of $F$‑isocrystals with $r$‑log decay. We first show that a rank one $F$‑isocrystal with $r$‑log decay is overconvergent if $r \lt 1$. Next, we establish a connection between slope gaps and the rate of log-decay of the slope filtration. The Drinfeld–Kedlaya theorem then follows from a patching argument.

中文翻译：

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$ F $-等晶体的斜率过滤和对数衰减

设$ k $为正特性的理想领域，并使$ X $为$ k $上的光滑不可约拟紧凑方案。Drinfeld-Kedlaya定理指出，对于在$ X $上不可约的$ F $-同晶，连续的一般斜率之间的距离以1为界。在本说明中，我们提供了该定理的新证明。我们的证明利用了具有$ r $ -log衰减的$ F $-异晶体的理论。我们首先表明，如果$ r \ lt 1 $，则具有$ r $ -log衰减的秩为$ F $的同晶将过度收敛。接下来，我们在边坡间隙和边坡过滤的对数衰减率之间建立联系。然后，Drinfeld-Kedlaya定理遵循一个修补性论证。