V. G. SPRINDŽUK

ON THE MEASURE OF THE SET OF \(S\)-NUMBERS IN A \(p\)-ADIC FIELD

(Presented by Academician I. M. Vinogradov on 5 IV 1963)

Let \(K(p)\) be the field of Hensel \(p\)-adic numbers, \(\omega \in K(p)\), and let \(|\omega|_p\) be the \(p\)-adic norm in \(K(p)\). In K. Mahler’s classification \((^1)\) of transcendental numbers from \(K(p)\), a number \(\omega\) is called an \(S\)-number if there exists a real number \(\upsilon>0\) such that

\[ |F(\omega)|_p>h^{-\upsilon},\qquad h>h_0(n,\omega), \]

for every integral polynomial \(F\) of degree \(n\) and height \(h\). Let \(\upsilon_n(\omega)\) be the exact lower bound of those \(\upsilon\) for which the indicated assertion is true.

D. Lock \((^2)\) proved that for almost all (in the sense of Turckstra measure \((^3)\)) numbers from \(K(p)\) the relations

\[ n+1\leq \upsilon_n(\omega)\leq 3n+1\qquad (n=1,2,\ldots) \]

hold. Later F. Kasch and B. Volkmann \((^4)\) obtained the inequalities

\[ \upsilon_n(\omega)\leq 2n-\tfrac12\qquad (n=3,4,\ldots), \]

as well as the equalities

\[ \upsilon_1(\omega)=2,\qquad \upsilon_2(\omega)=3 \]

for almost all numbers. At the same time, by analogy with Mahler’s known conjecture on real \(S\)-numbers, one may expect that

\[ \upsilon_n(\omega)=n+1\qquad (n=1,2,\ldots) \]

for almost all numbers from \(K(p)\).

We note that the arguments of Kasch and Volkmann \((^5)\) concerning the equality \(\upsilon_3(\omega)=4\) for almost all numbers are not correct.

Considerations analogous to those applied in the author’s papers \((^6,^7)\) lead to the following results.

Theorem 1. There exist numbers \(\upsilon_n\) such that

\[ \upsilon_n(\omega)=\upsilon_n\qquad (n=1,2,\ldots) \]

for almost all \(p\)-adic numbers \(\omega\). The quantities \(\upsilon_n\) satisfy the inequalities

\[ n+1\leq \upsilon_n\leq \tfrac54\bigl(n+\tfrac12\bigr)\qquad (n=3,4,5,6,7), \]

\[ n+1\leq \upsilon_n\leq \tfrac43 n\qquad (n=8,9,\ldots). \]

Theorem 2. The equality

\[ \upsilon_3(\omega)=4 \]

holds for almost all \(p\)-adic numbers \(\omega\).

For the proof of the last theorem, an estimate of the number of solutions of the Diophantine equation \(x^3=y^2+A\), found in \((^8)\), is essential.

Detailed proofs are contained in the author’s dissertation submitted to Leningrad State University named after A. A. Zhdanov.

Institute of Mathematics and Computing Technology
Academy of Sciences of the BSSR

Received
2 IV 1963

REFERENCES

\(^1\) K. Mahler, Mathematica, 3, 177 (1934–1935).
\(^2\) D. J. Lock, Vrije Univ. Amsterdam, Diss., 1947.
\(^3\) H. Turkstra, Vrije Univ. Amsterdam, Diss., 1936.
\(^4\) F. Kasch, B. Volkmann, Math. Zs., 72, 367 (1960).
\(^5\) F. Kasch, B. Volkmann, Math. Zs., 78, No. 2, 171 (1962).
\(^6\) V. G. Sprindžuk, Litovsk. matem. sborn., 2, No. 1, 129 (1962).
\(^7\) V. G. Sprindžuk, Litovsk. matem. sborn., 2, No. 2, 221 (1962).
\(^8\) V. G. Sprindžuk, Dokl. AN BSSR, 7, No. 1, 9 (1963).