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Historical statement

P. Erdős, Problems and results on diophantine approximations, Compositio Mathematica 16 (1964), 52–65, defines $N_n(u,v)$, for irrational $\alpha$, as the number of $1\le m\le n$ with $0\le u\le\{m\alpha\}<v\le1$ on printed p. 61. Equation (24), $N_n(u,v)=n(v-u)+O(1)$, is the theorem of Hecke and Ostrowski [22] there: bounded discrepancy when both endpoints are of the form $\{k\alpha\}$. On printed p. 62, Erdős explicitly reports the conjectured converse, attributed to himself and Szüsz: bounded discrepancy should force the two endpoints individually to be orbit points.

Thus the endpoint formulation of E0998 is present in this primary historical source. It is not solely a modern transcription error. The original problem statement is retained as a traceable variant.

Resolution of the literal and length formulations

The endpoint converse is false. Boris Alexeev's Lean theorem not_erdos_998, recorded on the Alexeev page, refutes it for $\alpha=\sqrt2/10$ with the interval of length $\alpha$ starting at $u=1/4$, neither endpoint lying in the orbit.

Kesten's later exact result, theorem_4, characterizes the length of a proper interval by $v-u=\{j\alpha\}$, for some integer $j$. It does not prove the false endpoint converse. A full reconstruction of Kesten's necessity proof remains separate from this historical-statement correction.