Problem 6

Given any real number a>1, construct a bounded infinite sequence

x_0,x_1,x_2,\ldots

such that

|x_n-x_m|\,|n-m|^a\ge1

for every pair of distinct n,m.

Step 3 of 4: Locate the first differing binary digit

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k=\nu_2(n-m),\qquad |n-m|\ge2^k

Detailed analysis

For distinct nonnegative integers $n,m$ , let $k=\nu_2(n-m)$ . Their binary digits below position $k$ agree, while the digit at position $k$ differs. Also $n-m$ is a nonzero multiple of $2^k$ , so $|n-m|\ge2^k$ . This remains valid when one index is 0, with $\nu_2(n-m)$ interpreted in the usual way.