| Summary: | In this paper, we explicitly describe all the elements of the sequence of fractional parts <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>a</mi><mrow><mi>f</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup><mo>/</mo><mi>n</mi><mo>}</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mo>…</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>∈</mo><mi mathvariant="double-struck">Z</mi><mo>[</mo><mi>x</mi><mo>]</mo></mrow></semantics></math></inline-formula> is a nonconstant polynomial with positive leading coefficient and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula> is an integer. We also show that each value <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>w</mi><mo>=</mo><mo>{</mo><msup><mi>a</mi><mrow><mi>f</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup><mo>/</mo><mi>n</mi><mo>}</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><msub><mi>n</mi><mi>f</mi></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>n</mi><mi>f</mi></msub></semantics></math></inline-formula> is the least positive integer such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>≥</mo><mi>n</mi><mo>/</mo><mn>2</mn></mrow></semantics></math></inline-formula> for every <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><msub><mi>n</mi><mi>f</mi></msub></mrow></semantics></math></inline-formula>, is attained by infinitely many terms of this sequence. These results combined with some earlier estimates on the gaps between two elements of a subgroup of the multiplicative group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="double-struck">Z</mi><mi>m</mi><mo>*</mo></msubsup></semantics></math></inline-formula> of the residue ring <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mi>m</mi></msub></semantics></math></inline-formula> imply that this sequence is everywhere dense in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula>. In the case when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>x</mi></mrow></semantics></math></inline-formula> this was first established by Cilleruelo et al. by a different method. More generally, we show that the sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>a</mi><mrow><mi>f</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup><mo>/</mo><msup><mi>n</mi><mi>d</mi></msup><mo stretchy="false">}</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mo>…</mo></mrow></semantics></math></inline-formula>, is everywhere dense in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula> if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>∈</mo><mi mathvariant="double-struck">Z</mi><mo>[</mo><mi>x</mi><mo>]</mo></mrow></semantics></math></inline-formula> is a nonconstant polynomial with positive leading coefficient and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula> are integers such that <i>d</i> has no prime divisors other than those of <i>a</i>. In particular, this implies that for any integers <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula> the sequence of fractional parts <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>a</mi><mi>n</mi></msup><mo>/</mo><mroot><mi>n</mi><mi>b</mi></mroot><mo stretchy="false">}</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mo>…</mo></mrow></semantics></math></inline-formula>, is everywhere dense in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula>.
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