| Summary: | In this paper, a theorem is obtained on the approximation in short intervals of a collection of analytic functions by shifts <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>ζ</mi><mrow><mo>(</mo><mi>s</mi><mo>+</mo><mi>i</mi><msubsup><mi>t</mi><mi>k</mi><msub><mi>α</mi><mn>1</mn></msub></msubsup><mo>)</mo></mrow><mo>,</mo><mo>…</mo><mo>,</mo><mi>ζ</mi><mrow><mo>(</mo><mi>s</mi><mo>+</mo><mi>i</mi><msubsup><mi>t</mi><mi>k</mi><msub><mi>α</mi><mi>r</mi></msub></msubsup><mo>)</mo></mrow><mo>)</mo></mrow></semantics></math></inline-formula> of the Riemann zeta function. Here, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msub><mi>t</mi><mi>k</mi></msub><mo>:</mo><mi>k</mi><mo>∈</mo><mi mathvariant="double-struck">N</mi><mo>}</mo></mrow></semantics></math></inline-formula> is the sequence of Gram numbers, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>α</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>α</mi><mi>r</mi></msub></mrow></semantics></math></inline-formula> are different positive numbers not exceeding 1. It is proved that the above set of shifts in the interval <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mi>N</mi><mo>,</mo><mi>N</mi><mo>+</mo><mi>M</mi><mo>]</mo></mrow></semantics></math></inline-formula>, here <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mo>=</mo><mi>o</mi><mo>(</mo><mi>N</mi><mo>)</mo></mrow></semantics></math></inline-formula> as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>→</mo><mo>∞</mo></mrow></semantics></math></inline-formula>, has a positive lower density. For the proof, a joint limit theorem in short intervals for weakly convergent probability measures is applied.
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