Integer Gap Questions+References

# Integer Gap Questions+References $\chi$=characteristic poly (0) What is the smallest gap in eigenvalues of an $n\times n$ integer matrix with small entries? Can it be $2^{-n^2}$? (1) Prob ($\chi(A)= \chi(B)$) is small ($2^{-n^2}$)? ($A,B$ symmetric or non, Bernoulli, 0/1, ...) (1') How many different $\chi$'s are there for bounded height $n\times n$ matrices? Update: we can construct $2^{n^2}$ matrices of size $2n\times 2n$ with entries $0,1,2$ that have distinct $\chi$'s. It now seems we can construct $h^{n^2}$ matrices of height $h$ and size $2n\times 2n$ with distinct and nearly irreducible characteristic polys. This means we get lots of distinct eigenvalues and can force a small gap if we allow a block sum matrix. (1'') Can we make these matrices symmetric? Can we make them irreducible? (2) Can graphs have exponentially small spectral gap? (3) Can we bound $|\lambda_2-\lambda_3|$ (for graphs, say)? (4) How many eigenvalues are there of bounded height $n\times n$ matrices? (5) How many integer matrices can be similar to each other? (6) How small can you make the entries in a matrix with given $\chi$? (6') How big do matrix entries have to be to get *most* char polys with small coeffs? (7) If two small integer matrices are similar, can we bound the entries of the conjugating matrix? (8) Smallest gap for real-rooted integer polys with small entries? (9) Given real-rooted integer poly, is there a symmetric integer matrix with this $\chi$? How big? # Root separation of polynomials https://www.jstor.org/stable/2005934 easiest argument giving the right asymptotics https://www.worldscientific.com/doi/pdf/10.1142/S1793042110003083?casa_token=vapmvNVfECcAAAAA%3A9KXlkpBWYrRk24E_TlK5oO-dhAfoYmPAFjoCPz-MiEwRNzB0t3A-gKmWdQQacsOGlpYIxyig5g& Bugeaud-Mignotte polynomial root separation survey http://irma.math.unistra.fr/~bugeaud/travaux/PolSurvRev1.pdf recent improvement of the constants in the above https://www.sciencedirect.com/science/article/pii/S0001870816305102 counting the number of integer polynomials with small discriminant https://londmathsoc.onlinelibrary.wiley.com/doi/pdfdirect/10.1112/blms/bdr085?casa_token=6v02vZXXTKQAAAAA%3AJZGPM3h-vMdFf1hhcc8nmV5fM1KRHzpCv5X3_TXvmn242J46Z1Iq_5e3w4b-j-1jM456UNsz0XrhUg Mahler'64: https://projecteuclid.org/journals/michigan-mathematical-journal/volume-11/issue-3/An-inequality-for-the-discriminant-of-a-polynomial/10.1307/mmj/1028999140.full Guting'61:https://projecteuclid.org/journals/michigan-mathematical-journal/volume-8/issue-2/Approximation-of-algebraic-numbers-by-algebraic-numbers/10.1307/mmj/1028998566.full Bugeaud-Mignotte Proc. Edinburgh: https://www.cambridge.org/core/services/aop-cambridge-core/content/view/287DEA74F09429BD971C047A936FB8C0/S0013091503000257a.pdf/on-the-distance-between-roots-of-integer-polynomials.pdf Distribution of conjugate roots: https://www.cambridge.org/core/services/aop-cambridge-core/content/view/98338A902AC2E079D613EB2FBB7B4EB1/S0010437X10004860a.pdf/distribution_of_close_conjugate_algebraic_numbers.pdf Mahler measure: https://en.wikipedia.org/wiki/Mahler_measure Gotze and Z: https://link.springer.com/content/pdf/10.1007/s10958-016-3139-9.pdf Bugeaud-Dujella: https://londmathsoc.onlinelibrary.wiley.com/doi/pdfdirect/10.1112/blms/bdr085?casa_token=27Oj0TB-qqsAAAAA:oNholWLAQGT4GMjT4xs2NicbB27_PU0qGNWgJ6ir8kuWszGvH9MwBVMFrgIhaHYlrzQZIBFmamD9Zw Mahler measure bound using Jensen's identity: https://carma.newcastle.edu.au/resources/mahler/docs/143.pdf ## Eigenvalues of Matrices https://apps.dtic.mil/sti/citations/ADA256570 tridiagonal example https://www.sciencedirect.com/science/article/pii/0024379581900033 $1/n^3$ lowerbound for graphs https://pdf.zlibcdn.com/dtoken/d18a9877a1f768f8660d5ee0c481b743/0509020.pdf another paper of Henry using interlacing to study gaps ## Random Matrix Theory Eberhard'20, irreducibility: https://arxiv.org/pdf/2008.01223.pdf Vu conjectures: https://arxiv.org/pdf/2005.02797.pdf Nguyen-Tao-Vu gaps:https://link.springer.com/article/10.1007/s00440-016-0693-5 Diaconis-Gamburd: https://www.emis.de/journals/EJC/Volume_11/PDF/v11i2r2.pdf Brezin-Hikami, considers products of determinants: https://arxiv.org/pdf/math-ph/9910005.pdf Bordenave-Chafai, some interesting combinatorics with Bernoulli moments: https://arxiv.org/pdf/2012.05602.pdf Kahn-Komlos-Szemeredi: https://www.ams.org/journals/jams/1995-08-01/S0894-0347-1995-1260107-2/S0894-0347-1995-1260107-2.pdf Tao-Vu CLT for determinant of Wigner: https://arxiv.org/pdf/1111.6300.pdf Tao-Vu, determinant is typically large: https://arxiv.org/pdf/math/0411095.pdf Vu talk: http://helper.ipam.ucla.edu/publications/cmaws1/cmaws1_8303.pdf ## Seidel Switching http://users.bestweb.net/~quenell/pubpdf/seidel.pdf Constructing cospectral graphs: http://users.cecs.anu.edu.au/~bdm/papers/GodsilMcKayCospectral.pdf A talk: http://www.math.ucsd.edu/~fan/teach/264/kenter/cospectral_talk.pdf