International Journal of Number Theory最新文献

{"title":"Explicit evaluation of triple convolution sums of the divisor functions","authors":"B. Ramakrishnan, Brundaban Sahu, Anup Kumar Singh","doi":"10.1142/s1793042124500544","DOIUrl":"https://doi.org/10.1142/s1793042124500544","url":null,"abstract":"<p>In this paper, we use the theory of modular forms and give a general method to obtain the convolution sums <disp-formula-group><span><math altimg=\"eq-00001.gif\" display=\"block\" overflow=\"scroll\"><mrow><msubsup><mrow><mi>W</mi></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></msubsup><mo stretchy=\"false\">(</mo><mi>n</mi><mo stretchy=\"false\">)</mo><mo>=</mo><munder><mrow><mo>∑</mo></mrow><mrow><mfrac linethickness=\"0\"><mrow><msub><mrow><mi>l</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>l</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>l</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>∈</mo><mi>ℕ</mi></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>l</mi></mrow><mrow><mn>1</mn></mrow></msub><mo stretchy=\"false\">+</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi>l</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy=\"false\">+</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>3</mn></mrow></msub><msub><mrow><mi>l</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>=</mo><mi>n</mi></mrow></mfrac></mrow></munder><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo stretchy=\"false\">(</mo><msub><mrow><mi>l</mi></mrow><mrow><mn>1</mn></mrow></msub><mo stretchy=\"false\">)</mo><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub><mo stretchy=\"false\">(</mo><msub><mrow><mi>l</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy=\"false\">)</mo><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></msub><mo stretchy=\"false\">(</mo><msub><mrow><mi>l</mi></mrow><mrow><mn>3</mn></mrow></msub><mo stretchy=\"false\">)</mo><mo>,</mo></mrow></math></span><span></span></disp-formula-group> for odd integers <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>≥</mo><mn>1</mn><mo>,</mo><mspace width=\"0.25em\"></mspace></math></span><span></span> and <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><mi>n</mi><mo>∈</mo><mi>ℕ</mi></math></span><span></span>, where <span><math altimg=\"eq-00004.gif\" display=\"inline\" overf","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140833094","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Near-miss identities and spinor genus classification of ternary quadratic forms with congruence conditions","authors":"Kush Singhal","doi":"10.1142/s1793042124500507","DOIUrl":"https://doi.org/10.1142/s1793042124500507","url":null,"abstract":"<p>In this paper, near-miss identities for the number of representations of some integral ternary quadratic forms with congruence conditions are found and proven. The genus and spinor genus of the corresponding lattice cosets are then classified. Finally, a complete genus and spinor genus classification for all conductor 2 lattice cosets of 2-adically unimodular lattices is given.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140833165","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Dense clusters of zeros near the zero-free region of ζ(s)","authors":"William D. Banks","doi":"10.1142/s1793042124500520","DOIUrl":"https://doi.org/10.1142/s1793042124500520","url":null,"abstract":"<p>The methods of Korobov and Vinogradov produce a zero-free region for the Riemann zeta function <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>ζ</mi><mo stretchy=\"false\">(</mo><mi>s</mi><mo stretchy=\"false\">)</mo></math></span><span></span> of the form <disp-formula-group><span><math altimg=\"eq-00004.gif\" display=\"block\" overflow=\"scroll\"><mrow><mi>σ</mi><mo>&gt;</mo><mn>1</mn><mo stretchy=\"false\">−</mo><mfrac><mrow><mi>c</mi></mrow><mrow><msup><mrow><mo stretchy=\"false\">(</mo><mo>log</mo><mi>τ</mi><mo stretchy=\"false\">)</mo></mrow><mrow><mn>2</mn><mo stretchy=\"false\">/</mo><mn>3</mn></mrow></msup><msup><mrow><mo stretchy=\"false\">(</mo><mo>log</mo><mo>log</mo><mi>τ</mi><mo stretchy=\"false\">)</mo></mrow><mrow><mn>1</mn><mo stretchy=\"false\">/</mo><mn>3</mn></mrow></msup></mrow></mfrac><mspace width=\"1em\"></mspace><mo stretchy=\"false\">(</mo><mi>τ</mi><mo>≔</mo><mo stretchy=\"false\">|</mo><mi>t</mi><mo stretchy=\"false\">|</mo><mo stretchy=\"false\">+</mo><mn>1</mn><mn>0</mn><mn>0</mn><mo stretchy=\"false\">)</mo><mo>.</mo></mrow></math></span><span></span></disp-formula-group> For many decades, the general shape of the zero-free region has not changed (although explicit known values for <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>c</mi></math></span><span></span> have improved over the years). In this paper, we show that if the zero-free region <i>cannot</i> be widened substantially, then there exist infinitely many distinct dense clusters of zeros of <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi>ζ</mi><mo stretchy=\"false\">(</mo><mi>s</mi><mo stretchy=\"false\">)</mo></math></span><span></span> lying close to the edge of the zero-free region. Our proof provides specific information about the location of these clusters and the number of zeros contained in them. To prove the result, we introduce and apply a variant of the original method of de la Vallée Poussin combined with ideas of Turán to control the real parts of power sums. We also prove similar results for <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>L</mi></math></span><span></span>-functions associated to <i>nonquadratic</i> Dirichlet characters <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mi>χ</mi></math></span><span></span> modulo <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><mi>q</mi><mo>≥</mo><mn>2</mn></math></span><span></span>.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140800385","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A note on logarithmic equidistribution","authors":"Gerold Schefer","doi":"10.1142/s1793042124500647","DOIUrl":"https://doi.org/10.1142/s1793042124500647","url":null,"abstract":"<p>For every algebraic number <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>κ</mi></math></span><span></span> on the unit circle which is not a root of unity we prove the existence of a strict sequence of algebraic numbers whose height tends to zero, such that the averages of the evaluation of <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>f</mi></mrow><mrow><mi>κ</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>z</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mo>log</mo><mspace width=\"-.17em\"></mspace><mo>|</mo><mi>z</mi><mo stretchy=\"false\">−</mo><mi>κ</mi><mo>|</mo></math></span><span></span> at the conjugates are essentially bounded from above by <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mo stretchy=\"false\">−</mo><mi>h</mi><mo stretchy=\"false\">(</mo><mi>κ</mi><mo stretchy=\"false\">)</mo></math></span><span></span>. This completes a characterization on functions <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>f</mi></mrow><mrow><mi>κ</mi></mrow></msub></math></span><span></span> initiated by Autissier and Baker–Masser, who cover the cases <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>κ</mi><mo>=</mo><mn>2</mn></math></span><span></span> and <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mo>|</mo><mi>κ</mi><mo>|</mo><mo>≠</mo><mn>1</mn></math></span><span></span>, respectively. Using the same ideas we also prove analogues in the <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>p</mi></math></span><span></span>-adic setting.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140800310","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Statistics for Iwasawa invariants of elliptic curves, II","authors":"Debanjana Kundu, Anwesh Ray","doi":"10.1142/s1793042124500556","DOIUrl":"https://doi.org/10.1142/s1793042124500556","url":null,"abstract":"<p>We study the average behavior of the Iwasawa invariants for Selmer groups of elliptic curves. These results lie at the intersection of arithmetic statistics and Iwasawa theory. We obtain lower bounds for the density of rational elliptic curves with prescribed Iwasawa invariants.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140625163","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Oscillations of Fourier coefficients of product of L-functions at integers in a sparse set","authors":"Babita, Mohit Tripathi, Lalit Vaishya","doi":"10.1142/s1793042124500854","DOIUrl":"https://doi.org/10.1142/s1793042124500854","url":null,"abstract":"<p>Let <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>f</mi></math></span><span></span> be a normalized Hecke eigenform of weight <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>k</mi></math></span><span></span> for the full modular group <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>S</mi><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy=\"false\">(</mo><mi>ℤ</mi><mo stretchy=\"false\">)</mo></math></span><span></span>. In this paper, we obtain the asymptotic of higher moments of general divisor functions associated to the Fourier coefficients of Rankin–Selberg <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>L</mi></math></span><span></span>-functions <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi>R</mi><mo stretchy=\"false\">(</mo><mi>s</mi><mo>,</mo><mi>f</mi><mo stretchy=\"false\">×</mo><mi>f</mi><mo stretchy=\"false\">)</mo></math></span><span></span>, supported at the integers represented by primitive integral positive-definite binary quadratic forms (reduced forms) of a fixed discriminant <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>D</mi><mo>&lt;</mo><mn>0</mn><mo>.</mo></math></span><span></span> We improve previous results in the case when the reduced form is given by <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mi mathvariant=\"cal\">𝒬</mi><mo stretchy=\"false\">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy=\"false\">)</mo><mo>=</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo stretchy=\"false\">+</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>.</mo></math></span><span></span></p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140599024","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Double and triple character sums and gaps between the elements of subgroups of finite fields","authors":"Jiankang Wang, Zhefeng Xu","doi":"10.1142/s1793042124500842","DOIUrl":"https://doi.org/10.1142/s1793042124500842","url":null,"abstract":"<p>For an odd prime <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>p</mi></math></span><span></span>, let <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>𝔽</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span><span></span> be the finite field of <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>p</mi></math></span><span></span> elements. The main purpose of this paper is to establish new results on gaps between the elements of multiplicative subgroups of finite fields. For any <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>∈</mo><msubsup><mrow><mi>𝔽</mi></mrow><mrow><mi>p</mi></mrow><mrow><mo stretchy=\"false\">∗</mo></mrow></msubsup></math></span><span></span>, we also obtain new upper bounds of the following double character sum <disp-formula-group><span><math altimg=\"eq-00005.gif\" display=\"block\" overflow=\"scroll\"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>χ</mi><mo>,</mo><msub><mrow><mi mathvariant=\"cal\">ℋ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant=\"cal\">ℋ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy=\"false\">)</mo><mo>=</mo><munder><mrow><mo>∑</mo></mrow><mrow><msub><mrow><mi>h</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∈</mo><msub><mrow><mi mathvariant=\"cal\">ℋ</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></munder><munder><mrow><mo>∑</mo></mrow><mrow><msub><mrow><mi>h</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∈</mo><msub><mrow><mi mathvariant=\"cal\">ℋ</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></munder><mi>χ</mi><mo stretchy=\"false\">(</mo><mi>a</mi><mo stretchy=\"false\">+</mo><mi>b</mi><msub><mrow><mi>h</mi></mrow><mrow><mn>1</mn></mrow></msub><mo stretchy=\"false\">+</mo><mi>c</mi><msub><mrow><mi>h</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy=\"false\">)</mo></mrow></math></span><span></span></disp-formula-group> and a triple character sum <disp-formula-group><span><math altimg=\"eq-00006.gif\" display=\"block\" overflow=\"scroll\"><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>χ</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><msub><mrow><mi mathvariant=\"cal\">ℋ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant=\"cal\">ℋ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mi mathvariant=\"cal\">𝒩</mi><mo stretchy=\"false\">)</mo><mo>=</mo><munder><mrow><mo>∑</mo></mrow><mrow><mi>x</mi><mo>∈</mo><mi mathvariant=\"cal\">𝒩</mi></mrow></munder><munder><mrow><mo>∑</mo></mrow><mrow><msub><mrow><mi>h</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∈</mo><msub><mrow><mi mathvariant=\"cal\">ℋ</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></munder><munder><mrow><mo>∑</mo></mrow><mrow><msub><mrow><mi>h</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∈</mo><msub><mrow><mi mathvariant=\"cal\">ℋ</mi>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140599021","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The transcendence of growth constants associated with polynomial recursions","authors":"Veekesh Kumar","doi":"10.1142/s1793042124500672","DOIUrl":"https://doi.org/10.1142/s1793042124500672","url":null,"abstract":"<p>Let <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>P</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>:</mo><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>d</mi></mrow></msub><msup><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msup><mo stretchy=\"false\">+</mo><mo>⋯</mo><mo stretchy=\"false\">+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><mi>ℚ</mi><mo stretchy=\"false\">[</mo><mi>x</mi><mo stretchy=\"false\">]</mo></math></span><span></span>, <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>a</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>&gt;</mo><mn>0</mn></math></span><span></span>, be a polynomial of degree <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>d</mi><mo>≥</mo><mn>2</mn></math></span><span></span>. Let <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mo stretchy=\"false\">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">)</mo></math></span><span></span> be a sequence of integers satisfying <disp-formula-group><span><math altimg=\"eq-00005.gif\" display=\"block\" overflow=\"scroll\"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi><mo stretchy=\"false\">+</mo><mn>1</mn></mrow></msub><mo>=</mo><mi>P</mi><mo stretchy=\"false\">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">)</mo><mspace width=\"1em\"></mspace><mstyle><mtext>for all </mtext></mstyle><mi>n</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mspace width=\"1em\"></mspace><mstyle><mtext>and</mtext></mstyle><mspace width=\"1em\"></mspace><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>→</mo><mi>∞</mi><mspace width=\"1em\"></mspace><mstyle><mtext>as </mtext></mstyle><mi>n</mi><mo>→</mo><mi>∞</mi><mo>.</mo></mrow></math></span><span></span></disp-formula-group></p><p>Set <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi>α</mi><mo>:</mo><mo>=</mo><msub><mrow><mo>lim</mo></mrow><mrow><mi>n</mi><mo>→</mo><mi>∞</mi></mrow></msub><msubsup><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow><mrow><msup><mrow><mi>d</mi></mrow><mrow><mo stretchy=\"false\">−</mo><mi>n</mi></mrow></msup></mrow></msubsup></math></span><span></span>. Then, under the assumption <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><msubsup><mrow><mi>a</mi></mrow><mrow><mi>d</mi></mrow><mrow><mn>1</mn><mo stretchy=\"false\">/</mo><mo stretchy=\"false\">(</mo><mi>d</mi><mo stretchy=\"false\">−</mo><mn>1</mn><mo stretchy=\"false\">)</mo></mrow></msubsup><mo>∈</mo><mi>ℚ</mi></math></span><span></span>, in a recent result by [A. Dubickas, Transcendency of some constants related to integer sequences of polynomial iterations, <i>Ramanujan J.</i><b>57</b> (2022) 569–581], either <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mi>α</mi></math></span><span></span> is transcendental or <span><math altimg=\"eq-00009.","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140599455","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A note on Schmidt’s subspace type theorems for hypersurfaces in subgeneral position","authors":"Lei Shi, Qiming Yan","doi":"10.1142/s1793042124500490","DOIUrl":"https://doi.org/10.1142/s1793042124500490","url":null,"abstract":"<p>In this paper, motivated by Nochka weights and the replacing hypersurfaces technique, we give an improvement of Schmidt’s subspace type theorem for hypersurfaces which are located in subgeneral position.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140599264","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A new upper bound on Ruzsa’s numbers on the Erdős–Turán conjecture","authors":"Yuchen Ding, Lilu Zhao","doi":"10.1142/s179304212450074x","DOIUrl":"https://doi.org/10.1142/s179304212450074x","url":null,"abstract":"<p>In this paper, we show that the Ruzsa number <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>R</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span><span></span> is bounded by <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mn>1</mn><mn>9</mn><mn>2</mn></math></span><span></span> for any positive integer <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>m</mi></math></span><span></span>, which improves the prior bound <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>R</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>≤</mo><mn>2</mn><mn>8</mn><mn>8</mn></math></span><span></span> given by Chen in 2008.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140599268","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}